`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

int(e^(tan^(-1)x))/(1+x^(2))dx

If int1/(xsqrt(1-x^3))dx=alog|(sqrt(1-x^3)-1)/(sqrt(1-x^3)+1)|+b ,t h e nai se q u a l 1/3 (b) 2/3 (c) -1/3 (d0 -2/3

int(e^(tan^(-1)x))/(1+x^(2))dx :

Ifint_0^(npi)f(cos^2x)dx=kint_0^pif(c0s^2x)dx ,t h e nfin dt h ev a l u ekdot

Prove that int_0^1tan^(-1)(1/(1-x+x^2))dx=2int_0^1tan^(-1)x dxdot Hence or otherwise, evaluate the integral int_0^1tan^(-1)(1-x+x^2)dx

Ifx^2+y^2=1a n dP=(3x-4x^3)^2+(3y-4y^3)^2,t h e nPi se q u a lto

Ifint(3e^x-5e^(-1))/(4e^x+5e^(-x))dx=a x+bln(4e^x+5e^(-x))+C ,t h e n

If int_0^1e^-(x^2)dx=a , then find the value of int_0^1x^2e^-(x^2)dx in terms of a .

int e^(tan^-1x)(1+x+x^2) d(cot^-1x) is equal to

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___