Prove that `int_0^(pi/2)sin^2x dx=int_0^(pi/2)cos^2x dx=pi/4`

int_0^(pi/2) sin x dx

(i) int_0^(pi/2) sin^2 x dx (ii) int_0^(pi//2) cos^2 x dx

Prove that int_0^(pi//2)sin^2x/(sinx+cosx)dx=int_0^(pi//2)(cos^2x)/(sinx+cosx)dx .

int_0^(pi//2) x^2 cos 2x dx

Prove that: int_0^(pi//2) cos^4 x dx =(3pi)/16

(i) int_0^(pi//2) sin^2 dx =pi/4 (ii) int_0^(pi//2) cos^2x dx=pi/4 (iii) int_(-pi//4)^(pi//4) sin^2 x dx=pi/4-1/2 (iv) int_(-pi//4)^(pi//4) cos^2x dx=pi/4+1/2

Prove that: int_(0)^(pi//2) log (sin x) dx =int_(0)^(pi//2) log (cos x) dx =(-pi)/(2) log 2

Prove that: int_(0)^(pi//2) log (sin x) dx =int_(0)^(pi//2) log (cos x) dx =(-pi)/(2) log 2

int_0^(pi/2) sin x sin 2x dx