Question

$\int e^x{\left(\frac{x+2}{x+4}\right)}^{2}dx$ gives $=e^x.\frac{x}{x+k}.$ find k?

Answer 1

$I=\int e^x\frac{(x^2+4x+4)}{{(x+4)}^2}dx$
$=\int e^x[\frac{x(x+4)}{{(x+4)}^2}+\frac{4}{{(x+4)}^2}]dx$
$=\int e^x[\frac{x}{x+4}+\frac{4}{{(x+4)}^2}]dx$
Above is of the form $\int e^x[f\left(x\right)+f^{\prime }\left(x\right)]dx$
$\therefore I=e^{x}f\left(x\right)=e^x.\frac{x}{x+4}.$