Question

Integrate $\int \frac{2x+3}{5x^2+1}dx$

Answer 1

$\int \frac{2x+3}{5x^2+1}dx$

$=\frac{1}{5}\int \frac{10x+15}{5x^2+1}dx$

$=\frac{1}{5}\int \frac{10x}{5x^2+1}dx+3\int \frac{dx}{5x^2+1}$

$=\frac{1}{5}\int \frac{10x}{5x^2+1}dx+\frac{3}{5}\int \frac{dx}{x^2+\frac{1}{5}}$

putting $5x^2+1=t$

$10xdx=dt$

$=\frac{1}{5}\int \frac{dt}{t}+\frac{3}{5}\int \frac{dx}{x^2+{\left(\frac{1}{\sqrt{5}}\right)}^2}$

$=\frac{1}{5}\log \left|t\right|+\frac{3}{5}\times \frac{1}{\left(\frac{1}{\sqrt{5}}\right)}{\tan }^{-1}\frac{x}{\left(\frac{1}{\sqrt{5}}\right)}+c$

$=\frac{1}{5}\log \left|5x^2+1\right|+\frac{3\sqrt{5}}{5}{\tan }^{-1}\left(\sqrt{5}x\right)+c$