RD Sharma Solutions for Class 12-science Maths CBSE Chapter 19: Indefinite Integrals

Indefinite Integrals Exercise Ex. 19.1

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Indefinite Integrals Exercise Ex. 19.2

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Indefinite Integrals Exercise Ex. 19.3

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Solution 19

Indefinite Integrals Exercise Ex. 19.4

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Indefinite Integrals Exercise Ex. 19.5

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Indefinite Integrals Exercise Ex. 19.6

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Indefinite Integrals Exercise Ex. 19.7

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Indefinite Integrals Exercise Ex. 19.8

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Solution 52

Indefinite Integrals Exercise Ex. 19.9

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Solution 19

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Solution 28

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Solution 31

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Solution 33

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Solution 35

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Solution 37

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Solution 40

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Solution 48

Solution 49

Solution 50

Solution 51

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Solution 53

Solution 54

$L e t space tan to the power of minus 1 end exponent x equals t D i f f e r e n t i a t i n g space t h e space a b o v e space f u n c t i o n space w i t h space r e s p e c t space t o comma space w comma space w e space h a v e comma fraction numerator 1 over denominator 1 plus x squared end fraction d x equals d t rightwards double arrow integral fraction numerator e to the power of m tan to the power of minus 1 end exponent x end exponent over denominator 1 plus x squared end fraction equals integral e to the power of m t end exponent cross times d t rightwards double arrow integral fraction numerator e to the power of m tan to the power of minus 1 end exponent x end exponent over denominator 1 plus x squared end fraction equals e to the power of m t end exponent over m R e s u b s t i t u t i n g space t h e space v a l u e space o f space t space i n space t h e space a b o v e space s o l u t i o n comma space w e space h a v e comma space space space space space space rightwards double arrow integral fraction numerator e to the power of m tan to the power of minus 1 end exponent x end exponent over denominator 1 plus x squared end fraction equals e to the power of m tan to the power of minus 1 end exponent x end exponent over m plus C$

Solution 55

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Solution 72

Indefinite Integrals Exercise Ex. 19.10

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Solution 8

Indefinite Integrals Exercise Ex. 19.11

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Solution 8

$begin mathsize 12px style Let space iota equals integral cos e c to the power of 4 3 x d x. space T h e n iota equals integral cos e c squared 3 x space cos e c squared 3 d x equals integral open parentheses 1 plus c o t squared 3 x close parentheses cos e c squared 3 x d x equals integral open parentheses cos e c squared 3 x space plus space c o t squared 3 x space cos e c squared 3 x close parentheses d x rightwards double arrow iota equals integral cos e c squared 3 x d x space plus space integral c o t squared 3 x space cos e c squared 3 x d x Substituting space c o t space 3 x space equals t space a n d space cos e c t squared 3 x d x space equals negative d t space i n space 2 nd space integral comma space we space get iota equals integral cos e c squared 3 x d x space minus space integral t squared fraction numerator d t over denominator 3 end fraction equals fraction numerator negative 1 over denominator 3 end fraction c o t 3 x minus t cubed over 9 plus C equals fraction numerator negative 1 over denominator 3 end fraction c o t space 3 x minus 1 over 9 c o t cubed space 3 x plus c therefore iota equals fraction numerator negative 1 over denominator 3 end fraction c o t space 3 x space minus space fraction numerator c o t cubed 3 x over denominator 9 end fraction plus c end style$

Solution 9

$begin mathsize 12px style L e t space iota equals integral c o t to the power of n x space cos e c squared x d x. n not equal to negative 1......... open parentheses i close parentheses L e t space c o t space x equals t. space T h e n d open parentheses c o t x close parentheses equals d t rightwards double arrow negative cos e c squared x d x equals d t rightwards double arrow cos e c squared x d x equals negative d t P u t t i n g space c o t x space equals t space a n d space c os e c squared x d x equals negative d t space i n space e q u a t i o n space open parentheses i close parentheses comma space w e space g e t iota equals integral t to the power of n cross times open parentheses negative d t close parentheses equals negative fraction numerator t to the power of n plus 1 end exponent over denominator n plus 1 end fraction plus c rightwards double arrow equals negative fraction numerator open parentheses c o t x close parentheses to the power of n plus 1 end exponent over denominator n plus 1 end fraction plus c end style$

Solution 10

Solution 11

Solution 12

Indefinite Integrals Exercise Ex. 19.12

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Solution 13

Indefinite Integrals Exercise Ex. 19.14

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Indefinite Integrals Exercise Ex. 19.15

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Solution 5

Indefinite Integrals Exercise Ex. 19.16

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Indefinite Integrals Exercise Ex. 19.17

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Solution 9

Indefinite Integrals Exercise Ex. 19.18

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Solution 17

Indefinite Integrals Exercise Ex. 19.19

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Solution 12

Indefinite Integrals Exercise Ex. 19.20

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Solution 10

Indefinite Integrals Exercise Ex. 19.32

Solution 3

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Solution 12

Solution 13

Solution 14

Indefinite Integrals Exercise Ex. 19.21

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Solution 18

Indefinite Integrals Exercise Ex. 19.22

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Solution 11

Indefinite Integrals Exercise Ex. 19.23

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Solution 15

Indefinite Integrals Exercise Ex. 19.24

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Solution 10

Solution 11

Indefinite Integrals Exercise Ex. 19.25

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Solution 14

Solution 15

Solution 16

Solution 17

Solution 18

Solution 19

Solution 20

Solution 21

Solution 22

Solution 23

Solution 24

Solution 25

Solution 26

Solution 27

Solution 29

Solution 30

Solution 31

Solution 32

Solution 33

Solution 34

Solution 35

Solution 36

Solution 37

Solution 38

Solution 39

Solution 40

Solution 41

Solution 42

Solution 43

Solution 44

Solution 45

Solution 46

Solution 47

Solution 48

Solution 49

Solution 50

Solution 51

Solution 52

Solution 54

Solution 55

Solution 56

Solution 57

Solution 58

Solution 59

Solution 60

Solution 61

Indefinite Integrals Exercise Ex. 19.26

Solution 1

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Solution 7

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Solution 12

Solution 13

Solution 14

Solution 15

Solution 16

Solution 17

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Solution 19

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Solution 21

Solution 22

Solution 23

Solution 24

Indefinite Integrals Exercise Ex. 19.27

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Solution 6

Solution 8

Solution 9

Solution 10

Solution 11

Solution 12

Indefinite Integrals Exercise Ex. 19.28

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Solution 12

Solution 13

Solution 14

Solution 15

Solution 16

Indefinite Integrals Exercise Ex. 19.29

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Solution 9

Solution 10

Solution 11

Solution 12

Indefinite Integrals Exercise Ex. 19.30

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Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

Solution 12

Solution 13

Solution 15

Solution 16

Solution 17

Solution 18

Solution 19

Solution 20

Solution 21

Solution 22

Solution 23

Solution 24

Solution 25

Solution 26

Solution 27

Solution 28

Solution 29

Solution 30

Solution 31

Solution 32

Solution 33

Solution 34

Solution 35

Solution 36

Solution 37

Solution 38

Solution 39

Solution 40

Solution 41

Solution 42

Solution 43

Solution 44

Solution 45

Solution 46

Solution 47

Solution 48

Solution 49

Solution 50

Solution 51

Solution 52

Solution 53

Solution 54

Solution 55

Solution 59

Solution 60

Solution 61

Solution 62

Solution 63

Solution 64

Solution 65

Indefinite Integrals Exercise Ex. 19.31

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Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

Indefinite Integrals Exercise MCQ

Solution 1

$begin mathsize 12px style Correct space option colon thin space left parenthesis straight b right parenthesis straight I equals integral fraction numerator straight x over denominator 4 plus straight x to the power of 4 end fraction dx space Put space straight x squared equals straight t rightwards double arrow 2 xdx equals dt rightwards double arrow xdx equals dt over 2 straight I equals integral fraction numerator begin display style dt over 2 end style over denominator 4 plus straight t squared end fraction straight I equals 1 half tan to the power of negative 1 end exponent open parentheses straight t over 2 close parentheses plus straight C straight I equals 1 half tan to the power of negative 1 end exponent open parentheses straight x squared over 2 close parentheses plus straight C end style$

Solution 2

$begin mathsize 12px style Correct space option colon left parenthesis straight c right parenthesis straight I equals integral fraction numerator 1 over denominator cosx plus square root of 3 sinx end fraction dx straight I equals 1 half integral fraction numerator 1 over denominator begin display style cosx over 2 end style plus begin display style fraction numerator square root of 3 over denominator 2 end fraction end style sinx end fraction dx straight I equals 1 half integral fraction numerator 1 over denominator cos open parentheses straight x minus begin display style straight pi over 6 end style close parentheses end fraction dx straight I equals 1 half integral sec open parentheses straight x minus straight pi over 6 close parentheses dx straight I equals 1 half ln open vertical bar tan open parentheses straight x over 2 plus straight pi over 3 close parentheses close vertical bar plus straight C end style$

Solution 3

$begin mathsize 12px style Correct space option colon left parenthesis straight a right parenthesis straight I equals integral xsecx squared dx Put space straight x squared equals straight t rightwards double arrow straight x equals square root of straight t rightwards double arrow 2 xdx equals dt rightwards double arrow xdx equals dt over 2 straight I equals integral sect dt over 2 straight I equals 1 half log open vertical bar sect plus tant close vertical bar plus straight C straight I equals 1 half log open vertical bar sec straight x squared plus tan straight x squared close vertical bar plus straight C end style$

Solution 4

$begin mathsize 12px style Correct space option colon space left parenthesis straight a right parenthesis integral fraction numerator 1 over denominator 5 plus 4 sinx end fraction dx equals Atan to the power of negative 1 end exponent open parentheses Btan straight x over 2 plus 4 over 3 close parentheses plus straight C Put space tan straight x over 2 equals straight t rightwards double arrow straight x equals 2 tan to the power of negative 1 end exponent straight t rightwards double arrow dx equals fraction numerator 2 dt over denominator 1 plus straight t squared end fraction rightwards double arrow sinx equals fraction numerator 2 tan begin display style straight x over 2 end style over denominator 1 plus tan squared straight x over 2 end fraction equals fraction numerator 2 straight t over denominator 1 plus straight t squared end fraction integral fraction numerator 1 over denominator 5 plus 4 sinx end fraction dx equals integral fraction numerator begin display style fraction numerator 2 dt over denominator 1 plus straight t squared end fraction end style over denominator 5 plus 4 cross times fraction numerator 2 straight t over denominator 1 plus straight t squared end fraction end fraction equals integral fraction numerator 2 dt over denominator 5 straight t squared plus 8 straight t plus 5 end fraction equals 2 over 5 integral fraction numerator dt over denominator straight t squared plus begin display style 8 over 5 end style straight t plus 1 end fraction Using space completing space square space method we space get straight I equals 2 over 3 tan to the power of negative 1 end exponent open parentheses 5 over 3 tan straight x over 2 plus 4 over 3 close parentheses plus straight C straight A equals 2 over 3 space and space straight B equals 5 over 3 end style$

Solution 5

$begin mathsize 12px style Correct space option colon thin space left parenthesis straight a right parenthesis straight I equals integral straight x to the power of sinx open parentheses sinx over straight x plus cosxlogx close parentheses dx Put space straight x to the power of sinx equals straight t taking space log space on space both space sides comma logt equals sinxlogx 1 over straight t dt equals sinx over straight x plus cosxlogx rightwards double arrow straight I equals integral straight t cross times dt over straight t straight I equals straight t plus straight C straight I equals straight x to the power of sinx plus straight C end style$

Solution 6

$begin mathsize 12px style Correct space option colon space left parenthesis straight b right parenthesis integral fraction numerator 1 over denominator 1 plus open parentheses log subscript straight e straight x close parentheses squared end fraction straight d open parentheses log subscript straight e straight x close parentheses Put space log subscript straight e straight x equals straight t integral fraction numerator dt over denominator 1 plus straight t squared end fraction equals tan to the power of negative 1 end exponent straight t plus straight C equals tan to the power of negative 1 end exponent open parentheses space log subscript straight e straight x close parentheses plus straight C end style$

Solution 7

$begin mathsize 12px style Correct space option colon thin space left parenthesis straight c right parenthesis integral fraction numerator cos 8 straight x plus 1 over denominator tan 2 straight x minus cot 2 straight x end fraction dx equals integral fraction numerator 2 cos squared 4 straight x over denominator begin display style fraction numerator sin 2 straight x over denominator cos 2 straight x end fraction end style minus begin display style fraction numerator cos 2 straight x over denominator sin 2 straight x end fraction end style end fraction dx equals integral fraction numerator 2 cos squared 4 straight x over denominator sin squared 2 straight x minus cos squared 2 straight x end fraction cross times sin 2 xcos 2 xdx equals integral negative fraction numerator cos squared 4 xsin 4 straight x over denominator cos 4 straight x end fraction dx equals fraction numerator negative 1 over denominator 2 end fraction integral sin 8 xdx equals fraction numerator cos 8 straight x over denominator 16 end fraction plus straight C straight a equals 1 over 16 end style$

Solution 8

$begin mathsize 12px style Correct space option colon left parenthesis straight a right parenthesis integral fraction numerator sin to the power of 8 straight x minus cos to the power of 8 straight x over denominator 1 minus 2 sin squared xcos squared straight x end fraction dx integral fraction numerator open parentheses sin to the power of 4 straight x plus cos to the power of 4 straight x close parentheses open parentheses sin to the power of 4 straight x minus cos to the power of 4 straight x close parentheses over denominator 1 minus 2 sin squared xcos squared straight x end fraction dx integral fraction numerator open parentheses sin to the power of 4 straight x plus cos to the power of 4 straight x close parentheses open parentheses sin squared straight x plus cos squared straight x close parentheses open parentheses sin squared straight x minus cos squared straight x close parentheses over denominator open parentheses sin squared straight x plus cos squared straight x close parentheses squared minus 2 sin squared xcos squared straight x end fraction dx integral fraction numerator open parentheses sin to the power of 4 straight x plus cos to the power of 4 straight x close parentheses open parentheses sin squared straight x minus cos squared straight x close parentheses over denominator open parentheses sin to the power of 4 straight x plus cos to the power of 4 straight x close parentheses end fraction dx integral negative cos 2 xdx fraction numerator negative sin 2 straight x over denominator 2 end fraction plus straight C rightwards double arrow straight a equals fraction numerator negative 1 over denominator 2 end fraction end style$

Solution 9

$begin mathsize 12px style Correct space option colon space left parenthesis straight a right parenthesis integral open parentheses straight x minus 1 close parentheses straight e to the power of negative straight x end exponent dx equals open parentheses straight x minus 1 close parentheses integral straight e to the power of negative straight x end exponent dx minus integral open parentheses open square brackets fraction numerator straight d open parentheses straight x minus 1 close parentheses over denominator dx end fraction close square brackets integral straight e to the power of negative straight x end exponent dx close parentheses dx equals open parentheses straight x minus 1 close parentheses fraction numerator straight e to the power of negative straight x end exponent over denominator negative 1 end fraction minus integral fraction numerator straight e to the power of negative straight x end exponent over denominator negative 1 end fraction dx equals negative open parentheses straight x minus 1 close parentheses straight e to the power of negative straight x end exponent plus fraction numerator straight e to the power of negative straight x end exponent over denominator negative 1 end fraction plus straight C equals negative xe to the power of negative straight x end exponent plus straight e to the power of negative straight x end exponent minus straight e to the power of negative straight x end exponent plus straight C equals negative xe to the power of negative straight x end exponent plus straight C end style$

Solution 10

$begin mathsize 12px style Correct space option colon thin space left parenthesis straight a right parenthesis straight I equals integral 2 to the power of begin display style 1 over straight x end style end exponent over straight x squared dx Put space 1 over straight x equals straight t rightwards double arrow fraction numerator negative 1 over denominator straight x squared end fraction dx equals dt rightwards double arrow 1 over straight x squared dx equals negative dt straight I equals integral 2 to the power of straight t open parentheses negative dt close parentheses straight I equals fraction numerator negative 2 to the power of straight t over denominator log subscript straight e 2 end fraction plus straight C rightwards double arrow straight I equals fraction numerator negative 2 to the power of begin display style 1 over straight x end style end exponent over denominator log subscript straight e 2 end fraction plus straight C straight k equals fraction numerator negative 1 over denominator log subscript straight e 2 end fraction end style$

Solution 11

$begin mathsize 12px style Correct space option colon space left parenthesis straight d right parenthesis straight I equals integral fraction numerator 1 over denominator 1 plus tanx end fraction dx equals integral fraction numerator cosx over denominator sinx plus cosx end fraction dx Numerator space can space be space written space as colon cosx equals straight A open parentheses sinx plus cosx close parentheses plus straight B fraction numerator straight d open parentheses sinx plus cosx close parentheses over denominator dx end fraction cosx equals open parentheses straight A minus straight B close parentheses sinx plus open parentheses straight A plus straight B close parentheses cosx rightwards double arrow straight A minus straight B equals 0 space and space straight A plus straight B equals 1 rightwards double arrow straight A equals 1 half equals straight B straight I equals integral fraction numerator open square brackets begin display style 1 half end style open parentheses sinx plus cosx close parentheses plus 1 half open parentheses cosx minus sinx close parentheses close square brackets dx over denominator sinx plus cosx end fraction straight I equals 1 half integral open parentheses 1 plus fraction numerator cosx minus sinx over denominator sinx plus cosx end fraction close parentheses dx straight I equals 1 half open square brackets 1 plus ln open parentheses sinx plus cosx close parentheses close square brackets plus straight C end style$

Solution 12

$begin mathsize 12px style Correct space option colon thin space left parenthesis straight d right parenthesis integral open vertical bar straight x close vertical bar cubed dx If space straight x greater than 0 rightwards double arrow integral straight x cubed dx equals straight x to the power of 4 over 4 plus straight C If space straight x less than 0 rightwards double arrow integral negative straight x cubed dx equals negative straight x to the power of 4 over 4 plus straight C end style$

Solution 13

$begin mathsize 12px style Correct space option colon thin space left parenthesis straight d right parenthesis straight I equals integral fraction numerator cos square root of straight x over denominator square root of straight x end fraction dx Put comma space square root of straight x space equals straight t fraction numerator 1 over denominator 2 square root of straight x end fraction dx equals dt rightwards double arrow fraction numerator 1 over denominator square root of straight x end fraction dx equals 2 dt straight I equals integral cost space 2 dt straight I equals 2 sint plus straight C equals 2 sin square root of straight x plus straight C end style$

Solution 14

$begin mathsize 12px style Correct space option colon space left parenthesis straight b right parenthesis straight I equals integral straight e to the power of straight x open parentheses 1 minus cotx plus cot squared straight x close parentheses dx straight I equals integral straight e to the power of straight x open parentheses 1 plus cot squared straight x minus cotx close parentheses dx straight I equals integral straight e to the power of straight x open parentheses cosec squared straight x minus cotx close parentheses dx Here comma space straight f left parenthesis straight x right parenthesis equals negative cotx rightwards double arrow straight f apostrophe left parenthesis straight x right parenthesis equals cosec squared straight x straight I equals negative straight e to the power of straight x cotx plus straight C end style$

Solution 15

$begin mathsize 12px style Correct space option colon thin space left parenthesis straight b right parenthesis straight I equals integral fraction numerator sin to the power of 6 xdx over denominator cos to the power of 8 straight x end fraction straight I equals integral tan to the power of 6 xsec squared xdx Put space tanx equals straight t rightwards double arrow sec squared xdx equals dt straight I equals integral straight t to the power of 6 dt straight I equals straight t to the power of 7 over 7 plus straight C straight I equals fraction numerator tan to the power of 7 straight x over denominator 7 end fraction plus straight C end style$

Solution 16

$begin mathsize 12px style Correct space option colon space left parenthesis straight a right parenthesis straight I equals integral fraction numerator dx over denominator 7 plus 5 cosx end fraction put space tan straight x over 2 equals straight t rightwards double arrow dx equals fraction numerator 2 dt over denominator 1 plus straight t squared end fraction rightwards double arrow cosx equals fraction numerator 1 minus tan squared begin display style straight x over 2 end style over denominator 1 plus tan squared straight x over 2 end fraction equals fraction numerator 1 minus straight t squared over denominator 1 plus straight t squared end fraction straight I equals integral fraction numerator fraction numerator 2 dt over denominator 1 plus straight t squared end fraction over denominator 7 plus 5 cross times fraction numerator 1 minus straight t squared over denominator 1 plus straight t squared end fraction end fraction rightwards double arrow straight I equals integral fraction numerator 1 over denominator straight t squared plus 6 end fraction dt straight I equals fraction numerator 1 over denominator square root of 6 end fraction tan to the power of negative 1 end exponent open parentheses fraction numerator straight t over denominator square root of 6 end fraction close parentheses plus straight C rightwards double arrow fraction numerator 1 over denominator square root of 6 end fraction tan to the power of negative 1 end exponent open parentheses fraction numerator tan begin display style straight x over 2 end style over denominator square root of 6 end fraction close parentheses plus straight C end style$

Solution 17

$begin mathsize 12px style Correct space option colon left parenthesis straight c right parenthesis integral fraction numerator dx over denominator 1 minus cosx minus sinx end fraction Put space straight t equals space tan straight x over 2 rightwards double arrow dx equals fraction numerator 2 dt over denominator 1 plus straight t squared end fraction cosx equals fraction numerator 1 minus tan squared straight x over 2 over denominator 1 plus tan squared straight x over 2 end fraction equals fraction numerator 1 minus straight t squared over denominator 1 plus straight t squared end fraction sinx equals fraction numerator 2 tan straight x over 2 over denominator 1 plus tan squared straight x over 2 end fraction equals fraction numerator 2 straight t over denominator 1 plus straight t squared end fraction put space in space the space question straight I equals integral fraction numerator begin display style fraction numerator 2 dt over denominator 1 plus straight t squared end fraction end style over denominator 1 minus fraction numerator 1 minus straight t squared over denominator 1 plus straight t squared end fraction minus fraction numerator 2 straight t over denominator 1 plus straight t squared end fraction end fraction straight I equals integral fraction numerator dt over denominator straight t squared minus straight t end fraction straight I equals integral fraction numerator dt over denominator straight t squared minus straight t plus begin display style 1 fourth end style minus begin display style 1 fourth end style end fraction rightwards double arrow straight I equals ln open vertical bar 1 minus cot straight x over 2 close vertical bar plus straight C end style$

Solution 18

$begin mathsize 12px style Correct space option colon thin space left parenthesis straight a right parenthesis straight I equals integral fraction numerator straight x plus 3 over denominator open parentheses straight x plus 4 close parentheses squared end fraction straight e to the power of straight x dx straight I equals integral open parentheses fraction numerator straight x plus 4 minus 1 over denominator open parentheses straight x plus 4 close parentheses squared end fraction close parentheses straight e to the power of straight x dx straight I equals integral open parentheses fraction numerator 1 over denominator straight x plus 4 end fraction minus 1 over open parentheses straight x plus 4 close parentheses squared close parentheses straight e to the power of straight x dx straight f left parenthesis straight x right parenthesis equals fraction numerator 1 over denominator straight x plus 4 end fraction rightwards double arrow straight f apostrophe left parenthesis straight x right parenthesis equals negative 1 over open parentheses straight x plus 4 close parentheses squared rightwards double arrow straight I equals fraction numerator straight e to the power of straight x over denominator straight x plus 4 end fraction plus straight C end style$

Solution 19

$begin mathsize 12px style Correct space option colon thin space left parenthesis straight c right parenthesis straight I equals integral fraction numerator sinx over denominator 3 plus 4 cos squared straight x end fraction dx Put space cosx equals straight t rightwards double arrow negative sinxdx equals dt rightwards double arrow sinxdx equals negative dt straight I equals integral fraction numerator negative dt over denominator 3 plus 4 straight t squared end fraction straight I equals fraction numerator negative 1 over denominator 2 square root of 3 end fraction tan to the power of negative 1 end exponent open parentheses fraction numerator 2 straight t over denominator square root of 3 end fraction close parentheses plus straight C straight I equals fraction numerator negative 1 over denominator 2 square root of 3 end fraction tan to the power of negative 1 end exponent open parentheses fraction numerator 2 cosx over denominator square root of 3 end fraction close parentheses plus straight C end style$

Solution 20

$begin mathsize 12px style Correct space option colon space left parenthesis straight b right parenthesis straight I equals integral straight e to the power of straight x open parentheses fraction numerator 1 minus sinx over denominator 1 minus cosx end fraction close parentheses dx straight I equals integral straight e to the power of straight x open parentheses fraction numerator 1 over denominator 1 minus cosx end fraction minus fraction numerator sinx over denominator 1 minus cosx end fraction close parentheses dx straight I equals integral straight e to the power of straight x open parentheses fraction numerator 1 over denominator 2 sin squared begin display style straight x over 2 end style end fraction minus fraction numerator 2 sin begin display style straight x over 2 end style cos straight x over 2 over denominator 2 sin squared straight x over 2 end fraction close parentheses dx straight I equals integral straight e to the power of straight x open parentheses fraction numerator cosec squared straight x over 2 over denominator 2 end fraction minus cot straight x over 2 close parentheses dx straight f left parenthesis straight x right parenthesis equals negative cot straight x over 2 rightwards double arrow straight f apostrophe left parenthesis straight x right parenthesis equals fraction numerator cosec squared straight x over 2 over denominator 2 end fraction straight I equals negative straight e to the power of straight x cot straight x over 2 plus straight C end style$

Solution 21

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Solution 22

$begin mathsize 12px style Correct space option colon left parenthesis straight c right parenthesis straight I equals integral fraction numerator straight e to the power of straight x open parentheses 1 plus straight x close parentheses over denominator cos squared open parentheses xe to the power of straight x close parentheses end fraction dx Put space xe to the power of straight x equals straight t rightwards double arrow straight e to the power of straight x open parentheses 1 plus straight x close parentheses dx equals dt straight I equals integral fraction numerator dt over denominator cos squared straight t end fraction straight I equals integral sec squared tdt straight I equals tant plus straight C straight I equals tan open parentheses xe to the power of straight x close parentheses plus straight C end style$

Solution 23

$begin mathsize 12px style Correct space option colon left parenthesis straight c right parenthesis straight I equals integral fraction numerator sin squared straight x over denominator cos to the power of 4 straight x end fraction dx straight I equals integral tan squared xsec squared xdx Put space tanx equals straight t rightwards double arrow sec squared xdx equals dt straight I equals integral straight t squared dt straight I equals straight t cubed over 3 plus straight C straight I equals fraction numerator tan cubed straight x over denominator 3 end fraction plus straight C end style$

Solution 24

$begin mathsize 12px style Correct space option colon left parenthesis straight a right parenthesis straight f left parenthesis straight x right parenthesis equals open parentheses 1 minus 1 over straight x squared close parentheses straight a to the power of straight x plus 1 over straight x end exponent rightwards double arrow integral straight f left parenthesis straight x right parenthesis dx equals integral open parentheses 1 minus 1 over straight x squared close parentheses straight a to the power of straight x plus 1 over straight x end exponent dx Put space straight x plus 1 over straight x equals straight t rightwards double arrow open parentheses 1 minus 1 over straight x squared close parentheses dx equals dt straight I equals integral straight a to the power of straight t dt straight I equals fraction numerator straight a to the power of straight t over denominator log subscript straight e straight a end fraction plus straight C straight I equals fraction numerator straight a to the power of straight x plus 1 over straight x end exponent over denominator log subscript straight e straight a end fraction plus straight C end style$

Solution 25

$begin mathsize 12px style Correct space option colon thin space left parenthesis straight d right parenthesis straight I equals integral fraction numerator 1 over denominator straight x plus xlogx end fraction dx straight I equals integral fraction numerator dx over denominator straight x open parentheses 1 plus logx close parentheses end fraction Put space 1 plus logx equals straight t rightwards double arrow 1 over straight x dx equals dt straight I equals integral 1 over straight t dt rightwards double arrow straight I equals log open vertical bar straight t close vertical bar plus straight C straight I equals log open parentheses 1 plus logx close parentheses plus straight C end style$

Solution 26

$begin mathsize 12px style Correct space option colon space left parenthesis straight d right parenthesis straight I equals integral square root of fraction numerator straight x over denominator 1 minus straight x end fraction end root dx straight I equals integral square root of fraction numerator straight x over denominator 1 minus straight x end fraction cross times straight x over straight x end root dx straight I equals integral fraction numerator xdx over denominator square root of straight x minus straight x squared end root end fraction Consider comma straight x equals straight A fraction numerator straight d open parentheses straight x minus straight x squared close parentheses over denominator dx end fraction plus straight B straight x equals straight A open parentheses 1 minus 2 straight x close parentheses plus straight B straight x equals negative 2 Ax plus straight A plus straight B minus 2 straight A equals 1 rightwards double arrow straight A equals fraction numerator negative 1 over denominator 2 end fraction rightwards double arrow straight A plus straight B equals 0 rightwards double arrow straight B equals 1 half straight I equals integral fraction numerator fraction numerator negative 1 over denominator 2 end fraction open parentheses 1 minus 2 straight x close parentheses plus 1 half over denominator square root of straight x minus straight x squared end root end fraction dx straight I equals integral open parentheses fraction numerator negative 1 over denominator 2 end fraction fraction numerator 1 minus 2 straight x over denominator square root of straight x minus straight x squared end root end fraction plus fraction numerator 1 over denominator 2 square root of straight x minus straight x squared end root end fraction close parentheses dx straight I equals fraction numerator negative 1 over denominator 2 end fraction cross times 2 square root of straight x minus straight x squared end root plus 1 half integral fraction numerator 1 over denominator square root of straight x minus straight x squared end root end fraction dx Second space term space after space completing space square space method space you space will space get space as straight I equals negative square root of straight x minus straight x squared end root plus sin to the power of negative 1 end exponent square root of straight x plus straight C end style$

Solution 27

$begin mathsize 12px style Correct space option colon left parenthesis straight a right parenthesis integral straight e to the power of straight x open curly brackets straight f open parentheses straight x close parentheses plus straight f apostrophe open parentheses straight x close parentheses close curly brackets dx equals straight e to the power of straight x straight f open parentheses straight x close parentheses plus straight C end style$

Solution 28

$begin mathsize 12px style Correct space option colon thin space left parenthesis straight d right parenthesis straight I equals integral fraction numerator sinx plus cosx over denominator square root of 1 minus sin 2 straight x end root end fraction dx straight I equals integral fraction numerator sinx plus cosx over denominator cosx minus sinx end fraction dx Put space cosx minus sinx equals straight t rightwards double arrow open parentheses sinx plus cosx close parentheses dx equals plus-or-minus dt rightwards double arrow open parentheses sinx plus cosx close parentheses dx equals plus-or-minus dt straight I equals plus-or-minus integral dt over straight t straight I equals plus-or-minus log open vertical bar straight t close vertical bar plus straight C straight I equals plus-or-minus log open vertical bar sinx minus cosx close vertical bar plus straight C end style$

Solution 29

$begin mathsize 12px style Correct space option colon space left parenthesis straight a right parenthesis integral xsinxdx equals negative xcosx plus straight alpha straight I equals integral xsinxdx straight I equals straight x integral sinxdx minus integral open parentheses fraction numerator d straight x over denominator d straight x end fraction integral sinxdx close parentheses dx straight I equals negative xcosx plus integral cosxdx straight I equals xcosx plus sinx plus straight C rightwards double arrow straight alpha equals sinx plus straight C end style$

Solution 30

$begin mathsize 12px style Correct space option colon space left parenthesis straight c right parenthesis straight I equals integral fraction numerator cos 2 straight x minus 1 over denominator cos 2 straight x plus 1 end fraction dx straight I equals negative integral fraction numerator 1 minus cos 2 straight x over denominator 1 plus cos 2 straight x end fraction dx straight I equals negative integral fraction numerator 2 sin squared straight x over denominator 2 cos squared straight x over 2 end fraction dx straight I equals negative integral tan squared xdx straight I equals negative integral open parentheses sec squared straight x minus 1 close parentheses dx straight I equals negative open parentheses tanx minus straight x close parentheses plus straight C straight I equals straight x minus tanx plus straight C end style$

Solution 31

$begin mathsize 12px style Correct space option colon left parenthesis straight a right parenthesis straight I equals integral fraction numerator cos 2 straight x minus cos 2 straight theta over denominator cosx minus cosθ end fraction dx straight I equals integral fraction numerator 2 cos squared straight x minus 1 minus open parentheses 2 cos squared straight theta minus 1 close parentheses over denominator cosx minus cosθ end fraction dx straight I equals integral fraction numerator 2 open parentheses cos squared straight x minus cos squared straight theta close parentheses over denominator cosx minus cosθ end fraction dx straight I equals 2 integral open parentheses cosx plus cosθ close parentheses dx straight I equals 2 open parentheses sinx plus xcosθ close parentheses plus straight C end style$

Solution 32

$begin mathsize 12px style Correct space option colon left parenthesis straight d right parenthesis straight I equals integral straight x to the power of 9 over open parentheses 4 straight x squared plus 1 close parentheses to the power of 6 dx straight I equals integral fraction numerator straight x to the power of 9 over denominator straight x to the power of 12 open parentheses 4 plus begin display style 1 over straight x squared end style close parentheses to the power of 6 end fraction dx straight I equals integral fraction numerator dx over denominator straight x cubed open parentheses 4 plus 1 over straight x squared close parentheses to the power of 6 end fraction Put space open parentheses 4 plus 1 over straight x squared close parentheses equals straight t rightwards double arrow fraction numerator negative 2 over denominator straight x cubed end fraction dx equals dt straight I equals 1 half integral negative straight t to the power of negative 6 end exponent dt straight I equals 1 half open parentheses fraction numerator negative straight t to the power of negative 5 end exponent over denominator negative 5 end fraction close parentheses plus straight C straight I equals 1 over 10 1 over straight t to the power of 5 plus straight C straight I equals 1 over 10 open parentheses 4 plus 1 over straight x squared close parentheses to the power of negative 5 end exponent plus straight C end style$

Solution 33

$begin mathsize 12px style Correct space option colon space left parenthesis straight d right parenthesis straight I equals integral fraction numerator straight x cubed over denominator square root of 1 plus straight x squared end root end fraction dx 1 plus straight x squared equals straight t rightwards double arrow 2 xdx equals dt rightwards double arrow xdx equals dt over 2 straight I equals integral fraction numerator straight x squared over denominator square root of 1 plus straight x squared end root end fraction xdx straight I equals integral fraction numerator straight t minus 1 over denominator square root of straight t end fraction dt over 2 straight I equals 1 half integral open parentheses square root of straight t minus straight t to the power of fraction numerator negative 1 over denominator 2 end fraction end exponent close parentheses dt straight I equals 1 half open parentheses 2 over 3 straight t to the power of 3 over 2 end exponent minus 2 square root of straight t close parentheses plus straight C straight I equals 1 third open parentheses 1 plus straight x squared close parentheses to the power of 3 over 2 end exponent minus square root of 1 plus straight x squared end root plus straight C straight a equals 1 third comma space straight b equals negative 1 end style$

Solution 34

$begin mathsize 12px style Correct space option colon left parenthesis straight d right parenthesis straight I equals integral fraction numerator straight x cubed over denominator straight x plus 1 end fraction dx straight I equals integral fraction numerator straight x cubed plus 1 minus 1 over denominator straight x plus 1 end fraction dx straight I equals integral fraction numerator open parentheses straight x plus 1 close parentheses open parentheses straight x squared minus straight x plus 1 close parentheses minus 1 over denominator straight x plus 1 end fraction dx straight I equals integral open parentheses straight x squared minus straight x plus 1 minus fraction numerator 1 over denominator straight x plus 1 end fraction close parentheses dx straight I equals straight x cubed over 3 minus straight x squared over 2 plus straight x minus log open vertical bar straight x plus 1 close vertical bar plus straight C end style$

Solution 35

$begin mathsize 12px style Correct space option colon left parenthesis straight c right parenthesis straight I equals integral fraction numerator 1 over denominator open parentheses straight x plus 2 close parentheses open parentheses straight x squared plus 1 close parentheses end fraction dx Consider comma fraction numerator 1 over denominator open parentheses straight x plus 2 close parentheses open parentheses straight x squared plus 1 close parentheses end fraction equals fraction numerator straight A over denominator straight x plus 2 end fraction plus fraction numerator Bx plus straight C over denominator open parentheses straight x squared plus 1 close parentheses end fraction 1 equals straight A open parentheses straight x squared plus 1 close parentheses plus open parentheses Bx plus straight C close parentheses open parentheses straight x plus 2 close parentheses Comaring space coefficeints space and space solving space it space simultaneously space we space get straight A equals 1 fifth comma space straight B equals negative 1 fifth comma space straight C equals 2 over 5 straight I equals integral open parentheses fraction numerator 1 over denominator 5 straight x plus 2 end fraction plus fraction numerator begin display style fraction numerator negative 1 over denominator 5 end fraction end style straight x plus begin display style 2 over 5 end style over denominator straight x squared plus 1 end fraction close parentheses dx Integrating space we space get space as 1 fifth log open vertical bar straight x plus 2 close vertical bar minus 1 over 10 log open vertical bar straight x squared plus 1 close vertical bar plus 2 over 5 tan to the power of negative 1 end exponent straight x plus straight C rightwards double arrow straight a equals negative 1 over 10 comma space straight b equals 2 over 5 end style$ $begin mathsize 12px style Correct space option colon left parenthesis straight c right parenthesis straight I equals integral fraction numerator 1 over denominator open parentheses straight x plus 2 close parentheses open parentheses straight x squared plus 1 close parentheses end fraction dx Consider comma fraction numerator 1 over denominator open parentheses straight x plus 2 close parentheses open parentheses straight x squared plus 1 close parentheses end fraction equals fraction numerator straight A over denominator straight x plus 2 end fraction plus fraction numerator Bx plus straight C over denominator open parentheses straight x squared plus 1 close parentheses end fraction 1 equals straight A open parentheses straight x squared plus 1 close parentheses plus open parentheses Bx plus straight C close parentheses open parentheses straight x plus 2 close parentheses Comaring space coefficeints space and space solving space it space simultaneously space we space get straight A equals 1 fifth comma space straight B equals negative 1 fifth comma space straight C equals 2 over 5 straight I equals integral open parentheses fraction numerator 1 over denominator 5 straight x plus 2 end fraction plus fraction numerator begin display style fraction numerator negative 1 over denominator 5 end fraction end style straight x plus begin display style 2 over 5 end style over denominator straight x squared plus 1 end fraction close parentheses dx Integrating space we space get space as 1 fifth log open vertical bar straight x plus 2 close vertical bar minus 1 over 10 log open vertical bar straight x squared plus 1 close vertical bar plus 2 over 5 tan to the power of negative 1 end exponent straight x plus straight C rightwards double arrow straight a equals negative 1 over 10 comma space straight b equals 2 over 5 end style$