Question

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

Answer 1

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

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

Let $5x^2+1=t$

$10xdx=dt$

$2xdx=\frac{dt}{5}$

$I={\int }_1^6\frac{dt}{5t}+3{\int }_0^1\frac{dx}{5x^2+1}$

$=\frac{1}{5}{\left[\ln t\right]}_1^6+\frac{3}{5}{\int }_0^1\frac{dx}{x^2+1/5}$

$=\frac{1}{5}(\ln 6-\ln 1)+\frac{3}{5}\left(\frac{1}{1/\sqrt{5}}\right){\left[{\tan }^{-1}\left(\frac{x}{1/\sqrt{5}}\right)\right]}_0^1$

$=\frac{\ln 6}{5}+\frac{3\sqrt{5}}{5}\tan {}^{-1}\left(\sqrt{5}\right)$