Solve the trigonometric integral $\int\sin\left(\frac{\pi x}{2}\right)dx$

Compute the integral

$\int\ln\left(x\right)dx$

Compute the integral

$\int\cos\left(4x\right)\cdot\cos\left(6x\right)dx$

Compute the integral

$\int\sec\left(5x\right)\cdot\tan\left(5x\right)dx$

Compute the integral

$\int\sin\left(3x\right)^2\cos\left(3x\right)dx$

Compute the integral

$\int\csc\left(x\right)dx$

Compute the integral

$\int\ln\left(x+1\right)dx$

Compute the integral

$\int\sin\left(4x\right)\cdot\cos\left(2x\right)dx$