Question

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

Answer 1

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

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

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

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

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substitute $u=x+1\to du=dx$

$=\int 1dx-2\int \frac{u-1}{u^2}du$

$=\int 1dx-2\int (\frac{1}{u}-\frac{1}{u^2})dx$

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