Mathematics_Today_-_October_2016

21. (log )
    (log )

x

x

− dx

+

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

⎭⎪

∫

1

1 2

2

is equal to

(a) xe
x

c

x

1 +^2

+ (b)

x

x

c

(log )^2 + 1

+

(c)

log

(log )

x

x

2 c

+ 1

+ (d) x

x

2 c

+ 1

+

22. Let fx xdx
    xx

()

()( )

=

+++

∫

2

11122

and f(0) = 0, then

the value of f(1) be

(a) log( 12 + ) (b) log( 12 )
4

+ −π

(c) log( 12 )
4

++π^ (d) none of these

23.∫cos−−^37 //xxdx⋅sin^117 =

(a) log | sin^47 / xc|+ (b)^4
7

tan^47 / xc+

(c) −^7 − +
4

tan^47 / xc (d) log|cos3/7x| + c

24. cos sin

/

θθθ

π

3

0

2

∫ d =

(a)^20
21

(b)^8

21

(c) −^20

21

(d) −^8

21

25. logx
    x

dx

a

b

∫ =

(a) log log
log

b

a

⎛

⎝⎜

⎞

⎠⎟

(b) log(ab) log b

a

⎛

⎝⎜

⎞

⎠⎟

(c)^1
2

log(ab) log b

a

⎛

⎝⎜

⎞

⎠⎟

(d)^1

2

log(ab) log a

b

⎛

⎝⎜

⎞

⎠⎟

26. ∫tan−^1 =
    0

1

xdx

(a) π
4

1

2

− log 2 (b) π−

1

2

log 2

(c) π
4

−log 2 (d) π−log 2

27. dx
    0 2 x

2

+

∫ =

cos

π/

(a)^1
3

1

3

tan−^1 ⎛

⎝⎜

⎞

⎠⎟

(b) 33 tan (−^1 )

(c)^2 ⎛

⎝⎜

⎞

⎠⎟

−

3

1

3

tan^1 (d) 23 tan (−^13 )

28. xdx
    ax

a 4

224

0 ()+

∫ =

(a)^1

16 4

1

a^33

⎛π−

⎝⎜

⎞

⎠⎟

(b)^1

16 4

1

a^33

⎛π+

⎝⎜

⎞

⎠⎟

(c)^1

16 4

1

3

a^3 ⎛π−

⎝⎜

⎞

⎠⎟ (d)

1

16 4

1

3

a^3 ⎛π+

⎝⎜

⎞

⎠⎟

29. sin cos
    sin

/ xx

x

+ dx

+

∫ =

0 916 2

π 4

(a)^1

20

log 3 (b) log3

(c)^1

20

log 5 (d) none of these

30. sin cos
    cos cos

/ xxdx

0 2 xx

2

++ 32

∫ =

π

(a) log^8

9

⎛

⎝⎜

⎞

⎠⎟ (b) log

9

8

⎛

⎝⎜

⎞

⎠⎟

(c) log(8 × 9) (d) none of these

31. The value of the integral

sinmxsinnxdxfor m n m n≠∈( , I),is

−

∫

π

π

(a) 0 (b) π (c) π

2

(d) 2π

32. dx
    012 − axa+^2

∫ =

cos

π

(a) π

21 ()−a^2

(b) π(1 – a^2 )

(c) π

1 −a^2

(d) none of these

33. The value of dx
    ∫ 1 +excos
    must be same as

(a)^1

1

1

2 12

1

−

−

+

⎛

⎝⎜

⎞

⎠⎟

− +

e

e

e

tan tanx c

(e lies between 0 and 1)

(b)^2

1

1

2 12

1

−

−

+

⎛

⎝⎜

⎞

⎠⎟

− +

e

e

e

tan tanx c,

(e lies between 0 and 1)