Integrals

6 0 |( 3x + 2x + x)| dx is equal to .

Let [ ] denote the greatest integer function and f(x) = n 1n3 k=1n [ k23x ]. Then 12 j=1 f(j) is equal to .

If 01 4 -1 (1 - 2x + 4x2 )dx = a -1(2) - b e(5), where a, b N, then (2a + b) is equal to .

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)

If ( 1x + 1x3 ) ( [23]3x-24 + x-26 ) dx = - 3( + 1) (3x + x ) + 1 + C, x > 0, ( , , Z), where C is the constant of integration, then + + is equal to .

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 f(x)+2f ( 1x )=x2+5 and 2g(x)-3g ( 12 )=x, x>0. If = 12 f(x) dx, and = 12 g(x) dx, then the value of 9 + is :

1. 0
2. 10
3. 1
4. 11

If ( 1+x2+x )10 ( 1+x2-x )9 dx= 1m ( ( 1+x2+x )n (n 1+x2-x ) )+C where C is the constant of integration and m, n N, then m+n is equal to .

Let f(x) = x3 3 - x2 dx. If 5f( 2) = -4, then f(1) is equal to

1. - 6 25
2. - 8 25
3. - 2 25
4. - 4 25

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 [.] denote the greatest integer function. If 0e3 [ 1ex-1 ] dx = - e 2, then 3 is equal to .

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

4 01 ( 1 3 + x2 + 1 + x2 )dx - 3 e( 3) is equal to :

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

The integral 80 0 4 ( + 9 + 16 2 )d is equal to :

1. 3 4
2. 4 3
3. 6 43
4. 2 3

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

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

Let f: R R be a twice differentiable function such that f(2) = 1. If F(x) = x f(x) for all x R, 02 x F'(x) dx = 6 and 02 x2 F''(x) dx = 40, then F'(2) + 02 F(x) dx is equal to:

1. 13
2. 11
3. 9
4. 15

If f(x) = 1x1/4 (1 + x1/4 )dx, f(0) = -6, then f(1) is equal to :

1. 4 ( e 2 - 2 )
2. e2 2 + 2
3. 2 - e2
4. 4 ( e 2 + 2 )

Let f be a real valued continuous function defined on the positive real axis such that g(x) = 0x t f(t) dt. If g (x3 ) = x6 + x7, then value of r=115 f (r3 ) is :

1. 270
2. 340
3. 310
4. 320