`IfI_n=int_0^pie^x(sin"x")^("n")dx ,t h e n(I_3)/(I_1)"is equal to"` `3/5` (b) `1/5` (c) 1 (d) `2/5`

IfI_n=int_0^1(1-x^5)^n dx ,t h e n(55)/7(I_(10))/(I_(11))i se q u a lto___

"I f"I_1=int_(-100)^(101)(dx)/((5+2x-2x^2)(1+e^(2-4x))) "and"I_2=int_(-100)^(101)(dx)/(5+2x-2x^2),t h e n(I_1)/(I_2)"i s" (a)2 (b) 1/2 (c) 1 (d) -1/2

L e tI_1=thetaint_0^1(e^xdx)/(1+x)a n dI_2thetaint_0^1(x^2dx)/(e^x^3(2-x^3))dott e h n(I_1)/(I_2)i se q u a lto 3/e (b) e/3 (c) 3e (d) 1/(3e)

Q. int_0^pi(e^(cos^2x)( cos^3(2n+1) x dx, n in I

If y=a x^(n+1)+b x^(-n),t h e nx^2(d^2y)/(dx^2) is equal to n(n-1)y (b) n(n+1)y (c) n y (d) n^2y

If int_1^2e^(x^2)dx=a ,t h e nint_e^(e^4)sqrt(1n x)dx is equal to (a) 2e^4-2e-a (b) 2e^4-e-a (c) 2e^4-e-2a (d) e^4-e-a

If y=cos^(-1)(cosx),t h e n(dy)/(dx) at x= (5pi)/4 is equal to (a)1 (b)-1 (c)0 (d)4

f(x) is a continuous function for all real values of x and satisfies int_n^(n+1)f(x)dx=(n^2)/2AAn in Idot Then int_(-3)^5f(|x|)dx is equal to (19)/2 (b) (35)/2 (c) (17)/2 (d) none of these

For x in R and a continuous function f, let I_1=int_(s in^2t)^(1+cos^2t)xf{x(2-x)}dxa n dI_2= int_(sin^2t)^(1+cos^2)xf{x(2-x)}dxdotT h e n(I_1)/(I_2) is -1 (b) 1 (c) 2 (d) 3

If f(x) is differentiable and int_0^(t^2)xf(x)dx=2/5t^5, then f(4/(25)) equals (a) 2/5 (b) -5/2 (c) 1 (d) 5/2