Question

Evaluate: $\int \frac{x^3}{(x+1{)}^2}dx$

Answer 1

$I=\int \frac{x^3}{(x+1{)}^2}dx$

Let $x+1=t\Rightarrow x=t-1$

$dx=dt$

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

$=\int \frac{t^3-1-3t^2+3t}{t^2}dt$

$=\int tdt-3\int dt-\int \frac{dt}{t^2}+\int \frac{3}{t}dt$

$=\frac{t^2}{2}-3t+\frac{1}{t}+3\ln |t|+9$

$=\frac{(x+1{)}^2}{2}-3x+\frac{1}{x+1}+3\ln |x+1|+C_2$

$=\frac{x^2}{2}-2x+\frac{1}{x+1}+3\ln |x+1|+C$