Integrals

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 24 0 4 ( |4x- 12 |+[2 x] )dx=2 + , where [ ] denotes the greatest integer function, then is equal to .