Question

Evaluate:$\int \frac{xdx}{x^2+2x+1}$

Answer 1

$\int \frac{xdx}{x^2+2x+1}$

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

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

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

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

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

$=\ln |x+1|+\frac{1}{x+1}+c$