`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)`

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___

int_(0)^(1)e^(2x)e^(e^(x) dx =)

Show that int_(0)^(1)(e^(x))/(1+e^(2x))dx=tan^(-1)(e)-pi/(4)

int(e^(3x)-e^(-3x))/(e^(3x)+e^(-3x))dx

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

Evaluate int_(0)^(1)e^(2-3x)dx

Ifint_0^1cot^(-1)(1-x+x^2)dx=lambdaint_0^1tan^(-1)x dx ,t h e nlambdai se q u a lto 1 (b) 2 (c) 3 (d) 4

Let f be a positive function. Let I_1=int_(1-k)^k xf([x(1-x)])dx , I_2=int_(1-k)^kf[x(1-x)]dx ,w h e r e2k-1> 0. T h e n(I_1)/(I_2)i s 2 (b) k (c) 1/2 (d) 1

Evaluate: int_0^1 e^(2-3x) dx