Question

Evaluate: $\int \frac{2x}{(x+5{)}^2}dx$

Answer 1

Let $I=\int \frac{2x}{(x+5{)}^2}dx$
$=\int \frac{2x+10-10}{(x+5{)}^2}dx$
$=2\int \frac{x+5}{(x+5{)}^2}dx-10\int \frac{1}{(x+5{)}^2}dx$
$=2\int \frac{1}{(x+5)}dx-10\int (x+5{)}^{-2}dx$
$=2\log |x+5|+\frac{10}{(x+5)}+c$
Where ${}^{\prime }{c}^{\prime }$ is a constant of Integration.