Prove that `int_(0)^(pi/2)sin2xlogtanxdx=0`.

Using the principle of mathematical induction to prove that int_(0)^(pi//2)(sin^2nx)/(sinx)dx=1+(1)/(3)+(1)/(5)+.....+(1)/(2n-1)

Evaluate int_(0)^(pi/2)logsinxdx

Evaluate the following integrals. int_(0)^(pi/2)sin2xlog(tanx)dx

Prove that int_(0)^(1)log((1)/(x)-1)dx=0 .

Prove that int_(0)^(pi)(x)/(1-cosalphasinx)dx=(pi(pi-alpha))/(sinalpha)

Evaluate the definite integrals int_(0)^(pi/4)sin2xdx

Prove the following int_(0)^(pi/4)2tan^(3)xdx=1-log2

Evaluate the definite integrals int_(0)^(pi/2)sin2xtan^(-1)(sinx)dx

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

Evaluate the definite integrals int_(0)^(pi/2)cos2xdx