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 .

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

If t 0 ( 01(3x+5)t dx ) 1t= 5e ( 85 ) 23, then is equal to .

If 2x2+5x+9 x2+x+1 dx =x x2+x+1+ x2+x+1 + e |x+ 12+ x2+x+1 |+C, where C is the constant of integration, then +2 is equal to .

Let n ( n n4 + 1 - 2n (n2 + 1 ) n4 + 1 . . + n n4 + 16 - 8n (n2 + 4 ) n4 + 16 + . . + n n4 + n4 - 2n n2 (n2 + n2 ) n4 + n4 ) be k, using only the principal values of the inverse trigonometric functions. Then k2 is equal to .

If 1 [5](x - 1)4(x + 3)6 dx = A ( x - 1 x + 3 )B + C, where C is the constant of integration, then the value of + + 20AB is .

Let rk = 01 (1 - x7 )k dx 01 (1 - x7 )k+1 dx, k N. Then the value of k=110 17 (rk - 1 ) 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 0 4 2x1+ x x dx= 1a e ( a3 )+ b 3, where a, b , then a+b is equal to .

If 5 x dx = x cosec x (cosec2 x + 32 ) + e | x2 | + C where , R and C is the constant of integration, then the value of 8( + ) 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:(0, ) and F(x)= 0x tf(t) dt. If F (x2 )=x4+x5, then r=112f (r2 ) is equal to .

If the integral 525 0 2 2x 112x (1+ 52x ) 12dx is equal to (n 2-64 ), then n is equal to .

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 .