Question

Solve $\int \frac{e^x\left(x^2+5x+7\right)}{{\left(x+3\right)}^2}dx$

Answer 1

$\int \frac{e^x\left(x^2+5x+7\right)}{{\left(x+3\right)}^2}dx$

let

$u^2={\left(x+3\right)}^2$

$⟹x^2+5x+7=u^2-u+1$

$\int \frac{e^{u-3}\left(u^2-u+1\right)}{{\left(u\right)}^2}du$

$=e^{-3}\int e^u\left(\frac{1}{u^2}-\frac{1}{u}+1\right)du$

$=e^{-3}\int e^{u}du+e^{-3}\int \left(\frac{1}{u}-\frac{1}{u^2}\right)e^{u}du$

$=e^{-3}{e}^u+\frac{e^{-3}}{u}$

$=e^{u-3}+\frac{e^{-3}}{u}$

Putting the value of u, we get

$=e^x+\frac{e^{-3}}{x+3}+c$