Integrals

The value of the integral 02 x (x2+x+1 ) ( x+1 ) ( x4+x2+1 ) dx is equal to :

1. 13 e (3-2 2 )
2. 23 e (4+ 2 )
3. 23 e (3+2 2 )
4. 13 e (1+6 2 )

If /6 /4 ( (x- 3 ) (x+ 3 )+1 )dx = e ( 3-1 ), then 9 2 is equal to .

The value of the integral - /4 /4 ( 32 4 x1+e x )dx is :

1. 4 +2
2. 3 +8
3. 3 +4
4. 4 +3

The value of the integral -11 ( x3+|x|+1x2+2|x|+1 )dx is equal to :

1. 3 e 2
2. 2 e 2
3. 5 e 3
4. 3 e 3

The value of the integral 0 e(x)x2+4 dx is:

1. e(2)2
2. e(2)4
3. 1+ e(2)
4. 2+ e(2)

The value of the integral 6 3 ( 4-cosec2 x 4 x )dx is :

1. 11 3
2. 16 3
3. 323 3
4. 643 3

Let (21-a+21+a ), f(a), (3a+3-a ) be in A.P. and be the minimum value of f(a). Then the value of the integral e( -1) e( ) dx (e2x-e-2x ) is :

1. 12 e ( 43 )
2. 14 e ( 43 )
3. 12 e ( 85 )
4. 14 e ( 85 )

Let f : [1, ) R be a differentiable function defined as f(x)= 1x f(t) dt +(1-x) ( e x-1 )+e. Then the value of f(f(1)) is :

1. (1+ee )
2. (1+e)
3. (1+e+ee )
4. 1+2e

Let -22 (| x|+[x x] )dx=2(3- 2)+ , where [ ] is the greatest integer function. Then ( 2 ) equals:

1. 1
2. 2
3. 4
4. 8

If =1 and =1+i 2, where i= -1 are two roots of the equation x3+ax2+bx+c=0, a, b, c R, then -11 (x3+ax2+bx+c )dx is equal to :

1. -2
2. -4
3. -8
4. -10

The integral 01 -1 (1+x+x2 )dx is equal to :

1. 2 -12+ 12 e ( 54 )+ 2
2. 2 -12+ 12 e ( 54 )- 2
3. 2 -12- 12 e ( 54 )+ 2
4. 2 -12- 12 e ( 54 )- 2

Let [.] denote the greatest integer function. Then the value of 03 ( ex+e-x[x]! )dx is :

1. e2+e3- 1e2- 1e3
2. 12 (e2+e3- 1e2- 1e3 )
3. e2+e3- 12e2- 12e3
4. 12 (e2+e3 )- 1e2- 1e3

If = 02 3 2 (x2+4 )dx+ 24 2x-4 dx, then 2 is equal to .

The value of 020 ( 4 x+ 4 x )dx is equal to:

1. 15 2
2. 25
3. 15
4. 25 2

Let f(x)= ( 16x+24x2+2x-15 )dx. If f(4)=14 e(3) and f(7)= e (2 3 ), , N, then + is equal to :

1. 31
2. 37
3. 39
4. 41

Let f be a polynomial function such that f(x2 + 1) = x4 + 5x2 + 2, for all x R. Then 03 f(x) dx is equal to

1. 332
2. 53
3. 272
4. 413

If ( 1 - 5 2 x 5 x 2 x )dx = f(x) + C, where C is the constant of integration, then f ( 6 ) - f ( 4 ) is equal to

1. 1 3(26 - 3)
2. 4 3(8 - 6)
3. 1 3(26 + 3)
4. 2 3(4 + 6)

The value of r=120 ( | ( 0r x| x|dx ) | ) is .

Let f(x) = dxx2/3 + 2x1/2, be such that f(0) = -26 + 24 e(2). If f(1) = a + b e(3), where a, b Z, then a + b is equal to :

1. -5
2. -11
3. -18
4. -26

Let [.] denote the greatest integer function. Then - 2 2 ( 12(3 + [x])3 + [ x] + [ x] ) dx is equal to :

1. 12 + 5
2. 11 + 2
3. 15 + 4
4. 13 + 1

Let f(t) = ( 1 - ( e t)1 - ( e t) ) dt, t > 1. If f (e /2 ) = -e /2 and f (e /4 ) = e /4, then equals

1. 1 + 2
2. -1 - 2 2
3. -1 - 2
4. -1 + 2

If f(x) satisfies the relation f(x) = ex + 01 (y + xex) f(y) dy, then e + f(0) is equal to .

Let f(x) = (2 - x2 ) ex ( 1 + x )(1 - x)3/2 dx. If f(0) = 0, then f ( 12 ) is equal to:

1. 2e - 1
2. 2e + 1
3. 3e - 1
4. 3e + 1

The value of the integral 24 5 24 dx1 + [3] 2x is :

1. 3
2. 18
3. 6
4. 12

Let I(x) = 3 dx(4x + 6) ( 4x2 + 8x + 3 ) and I(0) = 34 + 20. If I ( 12 ) = a 2b + c, where a, b, c N, (a, b) = 1, then a + b + c is equal to :

1. 30
2. 29
3. 28
4. 31

The number of elements in the set S = \x : x [0, 100] and 0x t2 (x - t) dt = x2 \ is

The value of - 2 2 ( 1[x] + 4 ) dx, where [ ] denotes the greatest integer function, is

1. 160(21 - 1)
2. 160( - 7)
3. 760( - 3)
4. 760(3 - 1)

If ( x) -112( x) -52 dx = - p1q1( x) 92 - p2q2( x) 52 - p3q3( x) 12 + p4q4( x) -32 + C, where pi and qi are positive integers with (pi, qi) = 1 for i = 1, 2, 3, 4 and C is the constant of integration, then 15p1p2p3p4q1q2q3q4 is equal to

Let [.] be the greatest integer function. If = 064 (x1/3 - [x1/3] )dx, then 1 0 ( 2 6 + 6 )d is equal to .

The value of - /6 /6 ( + 4x111 - (|x| + /6) )dx is equal to:

1. 8
2. 4
3. 2
4. 6