Prove the trigonometric identity $\frac{1}{2}\left(\cot\left(x\right)+\tan\left(x\right)\right)=\csc\left(2x\right)$

Compute the integral

$\int e^x\cos\left(x\right)dx$

Compute the integral

$\int xe^{2x}dx$

Compute the integral

$\int xe^{3x}dx$

Compute the integral

$\int3\cdot e^x\cdot xdx$

Compute the integral

$\int7\cdot e^{8x}\cdot xdx$

Compute the integral

$\int y^2 e^{2y}dy$

Compute the integral

$\int\sin\left(x\right)e^xdx$