Compute the integrals: `int_0^oof(x^n+x^(-n))logx(dx)/x`

Compute the integrals: int_(1/e)^e1/xsin(x-1/x)dx

evaluate the integrals int_(0)^(1)x(1-x)dx

evaluate the integrals int_(0)^(4)|x-1|dx

Evaluate the definite integrals int_(0)^(1)xe^(x^(2))dx

Let nge1, n in Z . The real number a in 0,1 that minimizes the integral int_(0)^(1)|x^(n)-a^(n)|dx is

If the value of the definite integral int_0^1(sin^(-1)sqrt(x))/(x^2-x+1)dx is (pi^2)/(sqrt(n)) (where n in N), then the value of n/(27) is

For any integer n , the integral int_0^pie^(cos^2x)cos^3(2n+1)xdx has the value

Prove that int_0^1tan^(-1)(1/(1-x+x^2))dx=2int_0^1tan^(-1)x dxdot Hence or otherwise, evaluate the integral int_0^1tan^(-1)(1-x+x^2)dx

If f(x) is continuous and int_(0)^(9)f(x)dx=4 , then the value of the integral int_(0)^(3)x.f(x^(2))dx is