Question

The value of $\int \frac{x^2+1}{(x+1{)}^2}dx$ is

Answer 1

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

Let $x+1=z$

$x=z-1$

$dx=dz$

$=\int \frac{(z-1{)}^2+1}{z^2}dz$

$=\int \frac{z^2+1-2z+1}{z^2}dz$

$=\int \frac{z^2-2z+2}{z^2}dz$

$=\int dz-2\int \frac{dz}{z}+2\int \frac{dz}{z^2}$

$=z-2\ln \mid z\mid +2\left(\frac{-1}{z}\right)+C$

$=(x+1)-2\ln \mid x+1\mid -\frac{2}{x+1}+C$