Question

Integrate:
$\int \frac{x^2}{(a+bx{)}^2}dx$

Answer 1

$I=\int \frac{x^2}{(a+bx{)}^2}dx$

Let $t=a+bx\Rightarrow bdx=dt$

$I=\int \frac{(t-a{)}^2}{b^2.t^2}\frac{dt}{b}=\frac{1}{b^3}\int \frac{t^2+a^2-2at}{t^2}dt$

$=\frac{1}{b^3}\int (1+\frac{a^2}{t^2}-\frac{2a}{t})dt$

$=\frac{1}{b^3}[t-\frac{a^2}{t}-2alog|t|]$

$I=\frac{1}{b^3}\left[\right(a+bx)-\frac{a^2}{(a+bx)}-2alog|bx+a|]+c$