Integrals

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)

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

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

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

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

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

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

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)

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

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

The integral -1 32 ( | 2 x ( x) | )dx is equal to:

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

Let f(x) be a positive function and I1 = - 121 2x f(2x(1 - 2x)) dx and I2 = -12 f(x(1 - x)) dx. Then the value of I2I1 is equal to

1. 12
2. 9
3. 6
4. 4

The integral 0 (x+3) x1+3 2 x dx is equal to

1. 3( +1)
2. 3 3( +6)
3. 3( +2)
4. 2 3( +4)

The value of -11 (1 + |x| - x ) ex + ( |x| - x ) e-xex + e-x dx is equal to

1. 1 + 2 23
2. 1 - 2 23
3. 2 + 2 23
4. 3 - 2 23

Let the domain of the function f(x) = 2 4 6 (3 + 4x - x2) be (a, b). If 0b-a [x2] dx = p - q - r, p, q, r N, (p, q, r) = 1, where [ ] is the greatest integer function, then p + q + r is equal to

1. 10
2. 11
3. 9
4. 8

The integral 0 8x dx4 2 x + 2 x is equal to

1. 2 2
2. 4 2
3. 2
4. 3 22

Let (a, b) be the point of intersection of the curve x2 = 2y and the straight line y - 2x - 6 = 0 in the second quadrant. Then the integral I = ab 9x21 + 5x dx is equal to :

1. 27
2. 18
3. 24
4. 21

If - 2 2 96x2 2 x(1 + ex) dx = ( 2 + ), , Z, then ( + )2 equals

1. 196
2. 100
3. 64
4. 144

The value of e2e4 1x ( e (( ex)2+1 )-1e (( ex)2+1 )-1+e ((6- ex)2+1 )-1 )dx is

1. 1
2. loge 2
3. e2
4. 2

If I= 0 2 32x 32x+ 32x dx, then 02I x x x 4x+ 4x dx equals :

1. 212
2. 24
3. 216
4. 28

The value of - 2y(1 + y)1 + 2 y dy is :

1. 2
2. 2
3. 22
4. 2 2

Let f(x)= cases-2, & -2 x 0 \ x-2, & 0<x 2 cases and h(x)=f(|x|)+|f(x)|. Then -22 h(x) dx is equal to:

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

If the value of the integral -11 x1 + 3x dx is 2 , then a value of is

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

The value of the integral 0 /4 x dx 4(2x)+ 4(2x) equals :

1. 2 28
2. 2 216
3. 2 232
4. 2 264

The value of 01 (2x3-3x2-x+1 ) 13dx is equal to :

1. -1
2. 2
3. 0
4. 1

If the value of the integral - 2 2 ( x2 x1+ x+ 1+ 2x1+e x2023 )dx= 4( +a)-2, then the value of a is

1. - 32
2. 3
3. 32
4. 2

If (a, b) be the orthocentre of the triangle whose vertices are (1,2),(2,3) and (3,1), and I1= ab x (4x-x2 )dx, I2= ab (4x-x2 )dx, then 36 I1I2 is equal to :

1. 80
2. 72
3. 66
4. 88