Integrals

Let y=f(x) be a thrice differentiable function in (-5,5). Let the tangents to the curve y=f(x) at (1,f(1)) and (3,f(3)) make angles /6 and /4, respectively with positive x-axis. If 27 13 ( (f'(t) )2+1 )f''(t) dt= + 3 where , are integers, then the value of + equals

1. 26
2. -16
3. 36
4. -14

Let f: R R be defined as f(x)=ae2x+bex+cx. If f(0)=-1, f' ( e2 )=21 and 0 e4(f(x)-cx) dx= 392, then the value of |a+b+c| equals

1. 16
2. 12
3. 8
4. 10

For x (- 2, 2 ), if y(x)= cosec x+ xcosec x x+ x 2x dx, and x ( 2 )-y(x)=0 then y ( 4 ) is equal to

1. - 1 2 -1 ( 1 2 )
2. -1 ( 1 2 )
3. 12 -1 ( 1 2 )
4. 1 2 -1 (- 12 )

If the value of the integral - 2 2 ( x2 x1+ x+ 1+ 2x1+e x2023 )dx= 4( +a)-2, then the value of a is

1. - 32
2. 3
3. 32
4. 2

If 32x+ 32x 3x 3x (x- ) dx =A x- +B - x+C, where C is the integration constant, then AB is equal to

1. 2
2. 8 cosec(2 )
3. 4 cosec(2 )
4. 4

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

If 01 1 3+x+ 1+x dx=a+b 2+c 3, where a, b, c are rational numbers, then 2a+3b-4c is equal to :

1. 10
2. 7
3. 4
4. 8

If (a, b) be the orthocentre of the triangle whose vertices are (1,2),(2,3) and (3,1), and I1= ab x (4x-x2 )dx, I2= ab (4x-x2 )dx, then 36 I1I2 is equal to :

1. 80
2. 72
3. 66
4. 88

The integral (x8-x2 )dx (x12+3x6+1 ) -1 (x3+ 1x3 ) is equal to :

1. e ( | -1 (x3+ 1x3 ) | )1/3+C
2. e ( | -1 (x3+ 1x3 ) | )+C
3. e ( | -1 (x3+ 1x3 ) | )1/2+C
4. e ( | -1 (x3+ 1x3 ) | )3+C

For 0<a<1, the value of the integral 0 dx1-2a x+a2 is :

1. 2 +a2
2. 2 -a2
3. 1-a2
4. 1+a2