∴ I= ∫

t

(^1) dt
3
1
34 / =
⎛
⎝
⎜
⎞
⎠
⎟+=
−

• ⎛
  ⎝⎜
  ⎞
  ⎠⎟ +
  1
  314
  4
  3
  1
  2
  t^1414
  c x
  x
  c
  / /
  /

15. (a) : I
    x

dx dx

xx

=

+

=

∫ ∫ +

1

12 sin^222 sin cos

=

+

=

+

∫∫

sec

tan

sec

tan

2

2

2

212

1

2 1

2

xdx

x

xdx

x

Put tanx = t ⇒ sec^2 xdx = dt, then

I dt

t

= t k

+

= ⎛

⎝⎜

⎞

⎠⎟

⎛

⎝⎜

⎞

⎠⎟

(^1) ∫ − +
2 1
2
1
2
1
(^21212)
1
/
tan
/
=+^1 −
2
tan (^12 tan )xk

16. (b) : Put tan–1(x^3 ) = z ⇒
    
    
    • 
      
    
    
    
    

(^1) ×=
1
6 3 2
x
xdx dz
Now, xx
x
dx zdz
213
1 6
1
3
tan (− )

• ∫∫=
  =^1 ⋅ =+−
  32
  1
  6
  z^2132
  (tan (xc))

17. (d) : I x
    x

==∫∫sin dx x⋅ xdx

cos

tan sec

3

5

(^232)
2
22
Putting tan2x = t and 2sec^22 xdx = dt, we get
I==∫tdt ⋅t+=cxc+
(^344)
2
1
24
1
8
(tan 2 )

18. (d) :Put 2xdxd=sinθθθ⇒ 2 =cos

⇒ =

−

Iddc∫∫cos ==+

sin

θ

θ

θθθ
1 2
⇒ I = sin–1(2x) + c

19. (b) : ∫fxdx gx() = ()(Given)

Now,Ifxdxfxdxd

dx

= ⋅ = − ⎧⎨ f x dx dx

⎩

⎫

⎬

⎭

−− −

∫ ∫ ∫ ∫

(^11) () () (^1) ()
=xf−−−−x−∫∫{}xd = −
dx
(^1111) () fxdxxfx xdfx() () { ()}
Let f–1(x)= t ⇒ x = f(t) and d{f–1(x)} = dt
∴Ixfx ftdtxfxgtxfxgfx= −−−−^1111 ()−∫ ()= () ()− = () {− ()}

20. (b) : sin
    sin cos

sin

sin cos

x

xx

dx x

xx

dx

−

=

∫∫−

1

2

2

= − ++

∫ −

1

2

(sin cos sin cos )

sin cos

xxxx

xx

dx

=++

−

⎛

⎝⎜

⎞

∫ ⎠⎟ =+ − +

1

2

1 1

2

sin cos

sin cos

xx [ log(sin cos )]

xx

dx x x x c

21. (b) : Put logxt dxedt= ⇒ = t
    log
    (log ) ( )

x

x

dx e

t

t

t

− t dt

+

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

⎭⎪

=

+

−

+

⎡

⎣

⎢

⎤

⎦

∫∫⎥

1

1

1

1

2

(^21)
2
222

• +=


• 
  

  e +
  t
  c x
  x
  c
  t
  1122 (log )

  
  

22. (b) : Let x = tanθ ⇒ dx = sec^2 θdθ

fx xdx

xx

() d

()( )

tan sec

sec ( sec )

=

+++

=

∫ ∫ +

2

22

22

111 2 1

θθθ

θθ

=

+

=

+

=

−

∫∫ +

tan

sec

sin

cos ( cos )

(cos)

cos ( cos

22 2

11

1

1

θθ

θ

θθ

θθ

θθ

θ

dd d

∫ θθ)

=∫∫(cos)− =∫ −

cos

1 θθ sec

θ

d θθ θdd

=++log(xx 1 21 ) tan− − xc+

Now,fc( )=++log( ) tan ( )0010− −^1 + ⇒^ c = 0 0

∴ (^) f( ) log( ) tan ( ) log( )1111 112
4
=++^21 − − =+−π

23. (c) : Ix xdx=∫cos−−^37 /(/)⋅(sin 2 37+ )
    =∫cos−−^37 //xx xdxsin^2 sin^37

=

⎛

⎝

⎜

⎞

⎠

⎟

∫∫cos =

cos

sin

cos

/ cot

/

/

ec^2 ec

37

37

2

37

x

x

x

dx xdx

x

Put cotx = t ⇒ –cosec^2 xdx = dt

I dt

t

=−∫ 37 =−^7 tc^47 +

/ 4

/ =−^7 − +

4

tan^47 / xc

24. (b) : LetId= ∫ cos sin

/

θθθ

π

3

0

2

Put t = cosθ ⇒ dt = –sinθdθ

It tdtttdt=−−∫^122 =∫^12 −^52

0

1

1

0

///() ( 1 )

⇒ (^) It t=⎡ −
⎣⎢
⎤
⎦⎥
(^2) =
3
2
7
8
21
32 72
0
1
//

25. (c) : LetI
    x

xdx

a

b

=∫^1 log

⇒ I xx

x

ab xdx

a

b

=[log .log ] −∫^1 log

⇒ 2 = ⇒ =^1 −

2

Ix I b a^222

a

[(log ) ]b [(log ) (log ) ]

⇒ =+ − = ⎛

⎝⎜

⎞

⎠⎟

Ibaba abb

a

1

2

1

2

[(log log )(log log )] log( ) log