Find the implicit derivative $\frac{d}{dx}\left(\ln\left(xy\right)=1\right)$

Evaluate the integral

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

Evaluate the integral

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

Evaluate the integral

$\int_0^{\infty}\left(\frac{x}{1+x^3}\right)dx$

Evaluate the integral

$\int_0^{\infty}\left(\frac{x}{x^3+1}\right)dx$

Evaluate the integral

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

Evaluate the integral

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

Evaluate the integral

$\int_1^2\frac{1}{x\cdot\left(x+1\right)}dx$