int_(0)^((pi)/(2))log(sin x)dx

int_(0)^((pi)/(2))log(sin2x)dx

Show that int_(0)^((pi)/(2))log(sin2x)dx=-(pi)/(2)(log2)

int_(0)^((pi)/(2))log(sinx)dx=int_(0)^((pi)/(2))log(cosx)dx=(pi)/(2)log.(1)/(2)

int_(0)^((pi)/(2))log(sinx)dx=int_(0)^((pi)/(2))log(cosx)dx=(pi)/(2)log.(1)/(2)

Using integral int_(0)^(-(pi)/(2))ln(sin x)dx=-int_(0)^( pi)ln(sec x)dx=-(pi)/(2)ln2 and int_(0)^((pi)/(2))ln(tan x)dx=0 and int_(0)^((pi)/(4))ln(1+tan x)dx=(pi)/(8)

Let u=int_(0)^(1)(ln(x+1))/(x^(2)+1)dx and v=int_(0)^((pi)/(2))ln(sin2x)dx, thenu=-(pi)/(2)ln2(b)4u+v=0u+4v=0 (d) u=(pi)/(8)ln2

By using the properties of definite integrals, evaluate the integrals int_(0)^((pi)/(2))(2log sin x-log sin2x)dx

Evaluate: int_(0)^((pi)/(2))log((4+3sin x)/(4+3cos x))dx

Prove that int_(0)^(pi//2)log (sinx)dx=int_(0)^(pi//2) log (cosx)dx=-(pi)/(2) log 2 .

Prove that int_(0)^(pi//2)log (sinx)dx=int_(0)^(pi//2) log (cosx)dx=-(pi)/(2) log 2 .