Integrals

If /6 /4 ( (x- 3 ) (x+ 3 )+1 )dx = e ( 3-1 ), then 9 2 is equal to .

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 .

The number of elements in the set S = \x : x [0, 100] and 0x t2 (x - t) dt = x2 \ is

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 .

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

Let [ ] denote the greatest integer function and f(x) = n 1n3 k=1n [ k23x ]. Then 12 j=1 f(j) is equal to .

If 01 4 -1 (1 - 2x + 4x2 )dx = a -1(2) - b e(5), where a, b N, then (2a + b) is equal to .

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 .

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