`Iff(x)=int_((x^2)/(16))^(x^2)(sinxsinsqrt(theta))/(1+cos^2sqrt(theta))dtheta,` then find the value of `f^(prime)(pi/2)dot`

If y(x)=int_((pi^(2))/(16))^(x^(2))(cos x cos sqrt(theta))/(1+sin^(2)sqrt(theta))d theta then find y'(pi)

If y(x)=int_((pi^(2))/(16))^(x^(2))(cos x cos sqrt(theta))/(1+sin^(2)sqrt(theta)), then f in d(dy)/(dx)atx=pi

If y(x)=\ int_(pi^2//16)^(x^2)(cos xdotcossqrt(theta))/(1+sin^2sqrt(theta))dotdtheta\ t h e n\ fin d(dy)/(dx)a t\ x=pi

If f(x)=int_(0)^((pi)/(2))(ln(1+x sin^(2)theta))/(sin^(2)theta)d theta,x>=0 then :

int(x)/(sqrt(1-x))dx=int(sin^(2)theta)/(sqrt(1-sin^(2)theta))*2sin theta cos theta d theta

If I_(1)=int_(0)^(1)(dx)/(e^(x)(1+x)) and I_(2)=int_(0)^(pi//4)(e^(tan^(7)theta)sintheta)/((2-tan^(2)theta)cos^(3)theta d theta ,then find the value of (l_(1))/(l_(2)) .

int_(0)^( pi)(sin theta+cos theta)/(sqrt(1+sin2 theta))d theta

If f(x)=(1)/(pi)int_(0)^((pi)/(2))(sin^(2)(n theta))/(sin^(2)theta)d(theta) then evaluate (f(15)+f(3))/(f(15)-f(9))

If f(x)=(1-x^(2))/(1+x^(2)) then show that f(tan theta)=cos2 theta