Integrals

If = 02 3 2 (x2+4 )dx+ 24 2x-4 dx, then 2 is equal to .

The value of r=120 ( | ( 0r x| x|dx ) | ) is .

If f(x) satisfies the relation f(x) = ex + 01 (y + xex) f(y) dy, then e + f(0) 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 .

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

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

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

Let [t] denote the largest integer less than or equal to t. If 03 ( [x2 ] + [ x22 ] )dx = a + b 2 - 3 - 5 + c 6 - 7, where a, b, c Z, then a + b + c is equal to .

If f(t) = 0 2x dx1 - 2 t 2 x, 0 < t < , then the value of 0 2 2 dtf(t) equals .

If - /2 /2 8 2 x dx (1+e x ) (1+ 4x )= + e(3+2 2), where , are integers, then 2+ 2 equals :

Let f : R R be a function defined by f(x)= 4x4x+2 and M= f(a)f(1-a)x 4(x(1-x))dx, N= f(a)f(1-a) 4(x(1-x))dx; a 12. If M= N, , N, then the least value of 2+ 2 is equal to .

| 120 3 0 x2 x x 4x+ 4xdx | is equal to .

The value of 9 09 [ 10xx+1 ]dx, where [t] denotes the greatest integer less than or equal to t, is .

If /6 /3 1- 2x dx= + 2+ 3, where , and are rational numbers, then 3 +4 - is equal to .

Let f(x)= 0x g(t) e ( 1-t1+t )dt, where g is a continuous odd function. If - /2 /2 (f(x)+ x2 x1+ex )dx= ( )2- , then is equal to .