Question

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

Answer 1

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

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

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

$I=\int e^x(\frac{1}{1+x^2}-\frac{2x}{(1+x^2{)}^2})dx$ ……..(i)

which is of the form $\int e^x\left(f\right(x)+f^{\prime }(x\left)\right)dx=e^x$ $f\left(x\right)+c$

where $f\left(x\right)=\frac{1}{1+x^2}$ such that $f^{\prime }\left(x\right)=\frac{-\left(2x\right)}{(1+x^2{)}^2}$

$\therefore I=e^x\left(\frac{1}{1+x^2}\right)+c$.