Question

Evaluate the following:
$\int \frac{2x+7}{(x-4{)}^2}dx$

Answer 1

$I=\int \frac{2x+7}{(x-4{)}^2}dx$

$I=\int \frac{2x-8+15}{(x-4{)}^2}dx=\int \frac{2x-8}{(x-4{)}^2}dx+\int \frac{15}{(x-4{)}^2}dx$

$I=\int \frac{2(x-4)}{(x-4{)}^2}dx+15\int \frac{1}{(x-4{)}^2}dx$

Let $x-4=t\Rightarrow dx=dt$

$I=2\int \frac{dx}{t}+15\int \frac{1}{t^2}dt$

$I=2\log \left(t\right)+15\frac{t^{-1}}{-1}$

$I=2log\left(t\right)-\frac{15}{t}$

$I=2log(x-4)-\frac{15}{(x-4)}+c$.