14 - indefinite integration

8/14/2019 14 - Indefinite Integration

1/11

PROBLEMS 14 - INDEFINITE INTEGRATION Page 1( Answers at the end of all problems )

Integrate the following functions with respect to x:

( 1 )

1

1x

x

x2

24

++

( 2 )1x

2x

2

6

++

( 3 )1x

x

2

4

+

( 4 ) 2

1

22

2

1

xlog -

( 5 )xsin-1

xsin ( 6 )

1e

1e

x

x

2

2 -

+

( 7 )xsinxcos

1

22( 8 )

xsinbxcosa

1

2222 + ( 9 ) xtan11+

( 10 )3x1x

1

-

( 11 )1x

1x

4

2 -

+ ( 12 ) xcos4xsin3xcosxsin

22 -

( 13 ))x2(

2xcos1


2/11


( 31 )4axx

1

-4( 32 )

)x4()1x(

1

--

( 33 )2

2

x

x25 -

( 34 )

)2ax0(

x2axx

1

2-< ( 35 ) cos x sin

22x

( 36 )2x2x8

1x3

--

+

( 37 ) x2

e3x

( 38 ) )2x()13x( - + ( 39 ) 5 xe 2x - 1

( 40 )

1x

x4

2

+ ( 41 ) x sinx cos 3x( 42 ) 2x2axx -

( 43 )2

1

x1

xsin -

+( 44 )

1xx

1

-

( 45 )

)0x(

x1

x1cos

2

21 --

( 48 )

x

2

2

e)1x(

1x

++

( 49 )xsin

5xsin ( 50 ) xtanx

12 - ( 51 ) x sin2x

( 52 ) x sec2x tan x ( 53 ) 2xsine 2x- ( 54 )

1x

1

-4

( 55 ) 2x

cos2

x ( 56 ) sec4

x

( 57 ) )4x()1x(

x

33

2

- +

( 58 ) e2x

sin 3x sin x ( 59 ) sec3

x( 60 )

)2x()23x(

x

-2 +


3/11


( 61 ))5x()2x(

x

22

3

+ ( 62 ) 15e6e1

x2x + ( 63 ) )4x()1x(1

2- +

( 64 )xsinxcos

xsinxcos11

11

--

---

+ ( 65 ) xcosxsin

( 66 ) x2

cos ( log x )

( 67 ))bx()ax(

x2222 + ( 68 ) xx

14xx

-

-3

2 +( 69 )

2)x(1)x(

x22

2

+

( 70 )

)2x0(

e

xcos1

xsin1

-- 2

x

/+ ( 71 )

)1x(x

1

3 +( 72 )

x4sin

xsin

( 73 ))xsin21(xcos

1+ ( 74 ) 2xsinxsin 1+ ( 75 ) xtan

( 76 )xa

x

- ( 77 )

1xx

1xx2

24

++

( 78 ) x1)x1( -

( 79 ) sin x cos x ( sin 2x +cos 2x ) ( 80 )

2xsin

xcosxsin + ( 81 )

22

22

xa

xax

-

+

( 82 )

xtan43

xtan

2+ ( 83 )2

3

2xx )ee(

1

-+( 84 )

34x)1x2(

1

+

( 85 )2

2

x1

x1

-

+ ( 86 )

2x1)x1(

1

-( 87 )

xsin32xcos

12+

( 88 ))4x()5x(

7x6

--

+

( 89 ) sin4

3x cos2

3x( 90 )

2xx0

1xsec

, /+


4/11


( 91 )

)ba(

xcosba

1

222 -> ( 92 ) xcotxtan + ( 93 )

xtanxtan1

xtan

2+

( 94 )2x2x1)x1(

1

- ( 95 )

1x)4x(

x

22 + ( 96 )17xx

1x

2

4

2

++

( 97 )xcosxsin

1

44 + ( 98 ) xcot ( 99 ) sin2x169cosxxsin+ +

( 100 )

xcosecxsec

1

+ ( 101 )

xx

xx

--

-

4e9e

6ee4 + ( 102 )

ax1

xa

-

-

( 103 ))1x(x

1

n + ( 104 ) secxxsin1+ ( 105 ) xtan

1-

( 106 )

4

1

x1

x1

++

( 107 )sinx1

x+ ( 108 ) xsin

( 109 ) tanx tan2x tan 3x ( 110 )

1xx

1

-43

( 111 ) x sin3

x

( 112 )xa

xtan 1- - ( 113 ) )5x()4x()3x(

)2x()1x(x ++ ( 114 ) x xa +

( 115 )xtan1

xtanxtan

3

3

++

( 116 )2

2

)cosxsinxx(

x

+ ( 117 ) n2n

x1

x

-

-

( 118 )222 )x1(x

1

+ ( 119 ) xa xsin - 1 + ( 120 ) sinx32cosx251

--


5/11


( 121 ))ax(sin

)ax(sin -+ ( 122 ) tanx34tanx23 ++ ( 123 )

x1

x1 -

+

( 124 )

)x(sinxsin

1

3 ( 125 ) xx - - ( 126 ) xcos4 xsin 22

+

( 127 )xtan1

14+ ( 128 ) 2)xlog(

1xloglog + ( 129 )

1x

x

-12

7

( 130 )

)cosx2sinx()2cosxsinx(

1

- + ( 131 ) )xx1(cot 21 -- + ( 132 )

cosxcos1)xcos1(

xsin

22

3

+

( 133 ) x ( tan 1

x )2 ( 134 )

4 53 )2x()1x(

1

- + ( 135 ) )3x()2x()1x(x

---

3

( 136 )

xcosxsincosxsinx)cosxsinx(

cosxxsin

22

-

+ ( 137 ) xsinx2cos

Answers

( Add constant C to all answers )

Answer:

[ 1 ] xtan1

+3

x -3

Answer:

[ 2 ] xtanx3

x

5

x 1--35

+

Answer:

[ 3 ] xtan+x3

x 1--3

Answer:

[ 4 ]2log

xsin 1-

Answer:

[ 5 ] sec x + tan x - x

Answer:

[ 6 ] log ( ex

+ e x

)


6/11


Answer:

[ 7 ] tan x - cot x

Answer:

[ 8 ]

xtan

a

btan

ab

1 1-

Answer:

[ 9 ] II xsinxcoslog2

1

2

x + Answer:

[ 10 ]

1+x1

1x1log

3

1

3

3

-

--

Answer:

[ 11 ]1+x2+x

1+x2xlog

2

2

22

1 -

Answer:

[ 12 ] II xcos4xsin3log 2214

1-

Answer:

[ 13 ] xsin2-

Answer:

[ 14 ] 2214

1)xtan( -

Answer:

[ 15 ] )4

x

8(tanlog)

2

x

4(sin22 ----

2

1

Answer:

[ 16 ] sin x + cos x

Answer:

[ 17 ]5

)xsin(4 2

5

Answer:

[ 18 ] xxcot3

xcot

5

xcot---

35

+

Answer:

[ 19 ] xsinxsin

1--

Answer:

[ 20 ]

)bx(sin

)ax(sinlog

)ba(sin

1

-

-

-

Answer:

[ 21 ]

)6xsin2xsin34xsin312x( --192

1 +

Answer:

[ 22 ] 23

1

3

2xsin -

Answer:

[ 23 ]

3x

tan

2

2

-+

Answer:

[ 24 ] )x1(log6x6x3x2 6

1

6

1

3

1

21

-- +

Answer:

[25]

21 x1xsin -- +

Answer:

[26]

24

xcos22

xtanlog2 +

Answer:

[27]

5x5

5x5log5x52

--

---

+

Answer:

[28]1x

2x2

-

-

Answer:

[29]

16

4xsin

4

x

4

2xsin--

Answer:

[30]

7

1x4tan

72

11x2xlog

4

3 12 - ++ Answer:

[31]2

21

2 a

xsec

a2

1 -

Answer:

[32]3

1xsin2

-- 1

Answer:

[33]5

xsin

x

x25 12

---

-


7/11


Answer:

[34]a

x2a

a

1 --

Answer:

[35] xsinxsin 5

5

43

3

4-

Answer:

[36]3

1xsin2x2x83 12 -----

+

Answer:

[37] 3x2

e27

2

9

2x

3

x-

+

Answer:

[38]

)2x()13x(32

56xlog

324

49)2x()13x(

12

5x6--- ++++

Answer:

[39])25log(

e5 12xx -

+

Answer:

[40]1x2x

1x2xlog

24

1

x2

1xtan

22

12

221 ---

+++

Answer:

[41]

4xcos8

x2xcos

4

x

8

2xsin

32

4xsin-- +

Answer:

[42]

2

3

2213

)x2ax(3

1x2ax)ax(

2

a

a

axsin

2

a----

-- +

Answer:

[43] )x1(log2

1xtanx 21 -- +

Answer:

[44]3

11x2tan

3

21xxlog

---- 1+

+

Answer:

[45]

+ 21 x1logxtanx2 ---

Answer:

[46]

2

5

22

7

2)x1(5

1

)x1(7

1-

++

Answer:

[47]22

22

ax

xaxxlog

-

--+

Answer:

[48]

+ 1x

1xe

-x

Answer:

[49]2

4xsin2xsinx +

Answer:

[50]

)]x1(logx[6

1

xtan3

x 2213

--- +

Answer:[51]

8

2xcos4

2xsinx

4

x

--

2

Answer:

[52]

2xtan

2xsecx -

2

Answer:

[53]

+ )4tan4x(cos17

11

2

e 1x

---

-

Answer:

[54] xtan2

1

1x

1xlog

4

1 1---+


8/11


Answer:

[55]

4)2log(

2log

2tan2xcos2

2log

2

2

11x1x

----

+

+

Answer:

[56]3

xtanxtan

3+

Answer:

[57]4x

1xlog

15

13

3 -

+ Answer:

[58] )4xsin24xcos(20

e)2xcos2xsin(

8

e-

2xx2 + Answer:

[59] xtanxseclogxtanxsec(2

1 + Answer:

[60]2

x3tan

67

123xlog

14

1

7

2xlog 12 --- +

Answer:

[61] )2x(log3

1)5x(log

6

5 22 - + Answer:

[62] + 2x3xx )12e()13e(elog - Answer:

[63]

2

xtan

10

1)4x(log

10

11xlog

5

1 12 ---- +

Answer:

[64]

211 xx2

)xsinxcos(12x

------

Answer:

[65] xsintanxsin1

xsin1log

2

1 1---

+

Answer:

[66] ])xlog(sin)xlog(cos3[10

x3 + Answer:

[67]

++

22

22

22 ax

bxlog

)ba(2

1

-

Answer:

[68]2

2

)1x(

)1x(xlog

-

+ Answer:

[69]2

xtan2xtan 11 --- +

Answer:

[70]2xsece 2

x

--

Answer:

[71]1x

xlog31 3

3

+ Answer:

[72]1xsin21xsin2log

241

1xsin1xsinlog

81 --- +

Answer:

[73] xsin21log3

2)xsin1(log

6

1)xsin1(log

2

1--- +


9/11


Answer:

[74] xcos21log3

2)xcos1(log

2

1)xcos1(log

6

1-- +

Answer:

[75]xtan21xtan

x2tan1xtanlog

22

1

xtan2

1xtantan

2

1 --- 1 +++

Answer:

[76] 23

2

1

)xa(3

2)xa(2a --- +

Answer:

[77] x2

x

3

x 23- +

Answer:

[78] 2

5

2

3

)x1(5

2)x1(

3

4--- +

Answer:

[79]16

4xcos

16

4xsin

4

x--

Answer:

[80] )xcosxsin(sin --1

Answer:

[81]

2

2

1244

axcosaxa

21 ---

Answer:

[82] xtan43tan 21- + Answer:

[83]1e3

2

3x

- + Answer:

[84]134x

134xlog

2

1 -

++

Answer:

[85]2

x1xxsin

2

3 21 ---

Answer:

[86]

x1

x1

+

-

Answer:

[87] )xtan2(tan2

1 1-

Answer:

[88]

209xxxlog34209xx6 --- 22

92 + Answer:

[89]144

6xsin

192

x12sin

16

x3

--

Answer:

[90] sin2sin2 2x1-

Answer:

[91]

xtan

ba

atan

baa

1

22

1

22 -

-

-

Answer:

[92] )xcosxsin(sin2 --1

Answer:

[93]

+

3

1xtan2tan

3

2x 1--

Answer:

[94]

+ x1 x2sin2

1 1-

Answer:

[95]3

1xtan

3

12

1- + Answer:

[96]

3x

1xtan

3

1 --2

1

Answer:

[97]

2

2xtantan

2

1 1-


10/11


Answer:

[98]1xcot2xcot

1xcot2xcotlog

22

1

xcot2

1xcottan

2

1 --

--- 1 ++

Answer:

[99])xcos(sinx45)xcosx(sin45

log401

-

Answer:

[100] +8

2

xtanlog

22

1)xcosxsin(

2

1--

Answer:[101]

-- 49elog36

35

2

3x 2x+ Answer:[102]

+ 22

3)ax1(

2

1)ax1()2a(ax1log)a1(

a

2-----

Answer:

[103]1x

xlog

n

1n

n

+ Answer:

[104] )cosxsinx(tan)cosxsinx(3

)cosxsinx(3log

32

1 1---

- +

Answer:

[105] xxtan)1(x --1+ Answer:

[106] 1xlog8x8x4x3

8xx

5

4 4

1

4

1

2

1

4

3

4

5

--- + Answer:

[107] x ( tanx secx) + log (1 + sinx )

Answer:

[108] [ ]xcoxxxsin2 - Answer:

[109] xseclog-2xseclog2

1-3xseclog

31

Answer:[110] 41

2

1 -- x

Answer:

[111]36

sin3x

12

cos3xx

4

cosx3x

4

sinx3-- +

Answer:

[112] 2xax2

1

a

xsin)

2

a2x( -- 1

Answer:

[113] 5xlog304xlog243xlog3x -- + Answer:

[114] axxlogaxax 2 +


11/11


Answer:

[115]3

1tanx2tan

3

11tanxxtanlog

6

1tanx1log

3

1 ---- 12 + Answer:

[116] sinxxxcoscosxxxsin -

+ Answer:

[117] 2n

1xsinn2 -

Answer:

[118] )x1(2

x

xtan2

3

x

1

2

1

--

-- + Answer:

[119] axa

xtan)ax( -- 1+

Answer:

[120]

6

2

xtan3tan

3

2--1

Answer:[121]

asinxsinsinxlogasinacos

xcoscosacos 22 -- +1-

Answer:[122]

sinx3cosx4log25

1

25

x18 + Answer:

[123] xcosx12xx 12 ---- + Answer:

[124] cotxsincossin

2- + Answer:

[125] -------- xsin)()x()x( 1 Answer:

[126] x5

tanx2tan

2

5--1

Answer:

[127]2xsin2

2xsin2log

24

1

2

x

-

++ Answer:

[128]xlog

xxloglogx -

Answer:

[129]

++3

12xtan

34

11xxlog

24

11xlog

12

1 414

84 ---

Answer:

[130]1tanx2

2tanxlog

5

1 -+ Answer:

[131] 1)(xtan1x

22xxlog

2

1)xx(1xcot --

--- 1

2

221 ++

++ Answer:

[132] xcos1xsectan 221- + Answer:

[133] 2111212

)xtan(2

1)xtan(seclogxtanx)xtan(

2

x ----- + Answer:

[134] 42x

1x

3

4 -+ Answer:

[135] 3xlog2

272xlog81xlog

2

1x ---- +

Answer:

[136] cosec 1( 1 + sin2x )

Answer: [137]x2tan-11

x2tan-11log

2

1

xtan12

xtan12log

2

1

--

--

-

2

2 +

14 - indefinite integration

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