Question

Evaluate: $\int \frac{(x+3)e^x}{(x+5{)}^3}dx$

Answer 1

$\Rightarrow \int \frac{x+3}{(x+5{)}^3}{e}^{x}dx$

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

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

$\int (\frac{1}{(x+5{)}^2}-\frac{2}{(x+5{)}^3})e^{x}dx$

$\int \left(f\right(x)+f^{\prime }(x\left)\right)e^{x}dx=f\left(x\right)e^x$

$Letf\left(x\right)=\frac{1}{(x+5{)}^2}then{f}^{\prime }\left(x\right)=\frac{-2}{(x+5{)}^3}$

So answer is , $\frac{e^x}{(x+5{)}^2}$