Integrals · Question 19

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 .