Show that: `int_0^(pi//2)f(sin2x)sinxdx=sqrt(2)int_0^(pi//4)f(cos2x)cosxdxdot`

Show that: int_(0)^( pi/2)f(sin2x)sin xdx=sqrt(2)int_(0)^( pi/4)f(cos2x)cos xdx

If the function f : [-1,1] to R is continuous and even, then show that int_(0)^(pi//2)f(cos2x)cosxdx=sqrt(2)int_(0)^(pi//4)f(sin2x)cosxdx .

int_(0)^(pi//4)x^(2)sinxdx

int_0^(pi/2) sqrt(cosx)sinxdx

int_0^(pi/2) x^2sinxdx

int_(0)^(pi//4)sqrt(1+cos2x)dx

If I_(1)=int_(0)^(pi//2)f(sin2x)sin x dx and I_(2)=int_(0)^(pi//4)f(cos2x)cosx dx , then I_(1)//I_(2) is equal to

If int_(0)^(pi/2) f ( sin2 x ) sin x dx = A int_(0)^(pi/4) f ( cos 2 x ) cos x dx then the value of A is ( sqrt2 = 1.41)

Show that (i) int_(0)^(pi//2)f(sinx) d x=int_(0)^(pi//2)f(cos x) d x (ii) int_(0)^(pi//2)f(tan x) d x=int_(0)^(pi//2)f(cot x) d x (iii) int_(0)^(pi//2)f(sin 2 x) sin xd x = int_(o)^(pi//2)f(sin 2x).cosx d x