Integrals · Question 43

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