Question

$\int \frac{e^x(x^3-x+2)}{{(1+x^2)}^2}dx=e^x.\frac{x+1}{x^2+1}.$ If this is true enter 1, else enter 0.

Answer 1

Let $I=\int \frac{e^x(x^3-x+2)}{{(1+x^2)}^2}dx$
$\frac{x^3-x+2}{{(x^2+1)}^2}=\frac{(x^2+1)(x+1)+(1-2x-x^2)}{{(x^2+1)}^2}$
$=\frac{x+1}{x^2+1}+\frac{1-2x-x^2}{{(x^2+1)}^2}=f\left(x\right)+f^{\prime }\left(x\right)$
$\therefore I=\int e^x[f\left(x\right)+f^{\prime }\left(x\right)]dx=e^x.\frac{x+1}{x^2+1}$