Prove the trigonometric identity $\frac{\sin\left(x\right)^2}{\cos\left(x\right)}+\cos\left(x\right)=\sec\left(x\right)$

Evaluate the integral

$\int_0^{\infty}\left(\frac{1}{1+x^2}\right)dx$

Evaluate the integral

$\int_{1}^{3}\cos\left(x\right)^2dx$

Evaluate the integral

$\int_{-5}^{5}\frac{1}{\sqrt{5-x}}dx$

Evaluate the integral

$\int_{2}^{4}\frac{1}{x^2-6x+5}dx$

Evaluate the integral

$\int_{2}^{4}\left(x^2+5x+6\right)dx$

Evaluate the integral

$\int_{2}^{5}\frac{1}{\left(x-1\right)\left(x+2\right)}dx$

Evaluate the integral

$\int_{1}^{2}\frac{1}{x\cdot\left(x+1\right)}dx$