Answer to Question #136210 in Calculus for qwerty

Question

\int \:\frac{xe^x}{\left(1+x^2\right)}dx

Answer 1

$\int \:\frac{xe^x}{\left(1+x\right)^2}dx =\int \:\frac{(x+1-1)e^x}{(1+x)^2}dx =\int \:\frac{e^x}{1+x}dx -\int \:\frac{e^x}{(1+x)^2}dx =$

$|u=\frac{1}{x+1}, dv=e^xdx, du=-\frac{1}{(1+x)^2}dx, v=e^x|$

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