Prove the trigonometric identity $\frac{\sin\left(x\right)}{\cos\left(x\right)}+\frac{\cos\left(x\right)}{1+\sin\left(x\right)}=\sec\left(x\right)$

Compute the integral

$\int\frac{x}{\sqrt{x^2+9}}dx$

Compute the integral

$\int\frac{dx}{x^2\sqrt{4-x^2}}$

Compute the integral

$\int\frac{1}{\sqrt{x^2-36}}dx$

Compute the integral

$\int\frac{1}{\sqrt{1-x^2}}dx$

Compute the integral

$\int\frac{x}{\sqrt{16-x^2}}dx$

Compute the integral

$\int\frac{1}{\sqrt{1-5x}}dx$

Compute the integral

$\int\frac{x^3}{\sqrt{x^2+4}}dx$