Question

$\underset{0}{\overset{2}{\int }}\frac{5x+2}{x^2+4}dx$

Answer 1

$\underset{0}{\overset{2}{\int }}\frac{5x+2}{x^2+4}dx$

$\int \frac{5x}{x^2+4}dx+\int \frac{2}{x^2+4}dx$

$5\int \frac{x}{x^2+4}dx+\int \frac{2}{x^2+4}dx$

$x^2+4=t$

$2xdx=dt$

$\frac{5}{2}\log \left(x^2+4\right)+2{\tan }^{-1}\left(x\right)\underset{0}{\overset{2}{\int }}$

$=\frac{5}{2}\log \left(8\right)-\frac{5}{2}\log \left(4\right)+\pi /4$