Integrals · Question 34

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 .