Question

Evaluate the following definite integral:

${\int }_0^1\frac{2x+3}{5x^2+1}dx$

Answer 1

$I={\int }_0^1\frac{2x+3}{5x^2+1}dx={\int }_0^1\frac{2x}{5x^2+1}dx+{\int }_0^1\frac{3}{5x^2+1}dx=\frac{1}{5}{\int }_0^1\frac{10x}{5x^2+1}dx+3{\int }_0^1\frac{1}{(\sqrt{5}x{)}^2+1}dx$

$\Rightarrow I=\frac{1}{5}\left[\log \right(5x^2+1){]}_0^1+\frac{5}{\sqrt{5}}{\left[{\tan }^{-1}\frac{\sqrt{5}x}{1}\right]}_0^1$

$\Rightarrow I=\frac{1}{5}(\log 6-\log 1)+\frac{3}{\sqrt{5}}{\tan }^{-1}\sqrt{5}=\frac{1}{5}\log 6+\frac{3}{\sqrt{5}}{\tan }^{-1}\sqrt{5}$