`Iff(x)=int_0^pi(tsintdt)/(sqrt(1+tan^2xsin^2t))for0

If f(x)=int_(0)^(pi)(t sin t dt)/(sqrt(1+tan^(2)xsin^(2)t)) for 0lt xlt (pi)/2 then

int_(0)^( pi/2)(dx)/(1+tan^(5)x)

int(xsin^-1x)/sqrt(1-x^2)dx

int_(0)^(pi//2)(1)/(1+sqrt(tan x))dx=

Find the mistake of the following evaluation of the integral I=int_(0)^( pi)(dx)/(1+2sin^(2)x)I=int_(0)^( pi)(dx)/(cos^(2)x+3sin^(2)x)=int_(0)^( pi)(sec^(2)xdx)/(1+3tan^(2)x)=(1)/(sqrt(3))[tan^(-1)(sqrt(3)tan x)]_(0)^( pi)=0

int_(0)^(2pi)sqrt(1+"sin"x/2)dx=

int_(0)^(pi//2)(1)/(sqrt(tan x)-sqrt(cot x))dx=

If f(x)=int_0^sinx cos^-1t dt+int_0^cosx sin^-1t dt, 0ltxltpi/2 , then f(pi/4)= (A) 0 (B) pi/sqrt(2) (C) 1 (D) pi/(2sqrt(2))