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

Evaluate the integral

$\int_2^{\infty}\left(\frac{1}{x^2+x-2}\right)dx$

Evaluate the integral

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

Find the integral

$\int\frac{1}{\left(x^4+1\right)}dx$

Find the integral

$\int\frac{dx}{x^2+7x-6}$

Find the integral

$\int\frac{dx}{x^2+7x+6}$

Evaluate the integral

$\int_2^{\infty}\left(\frac{1}{x^2+6x+5}\right)dx$

Evaluate the integral

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