Question

$\int e^{x} \left(\frac{x+2}{x+4}\right)^{2} dx = f\left(x\right) + $ arbitrary constant, then $f(x)$ =

• A. $\frac{xe^x}{x +4}$
• B. $\frac{e^x}{x +4}$
• C. $\frac{xe^x}{(x +4)^2}$
• D. $\frac{e^x}{(x +4)^2}$

Answer 1

Answer: (A)

Given, $\int e^{x}\left(\frac{x+2}{x+4}\right)^{2} d x=f(x)+c$
Now, $\int e^{x}\left(\frac{x+2}{x+4}\right)^{2} d x=\int e^{x}\left(\frac{x^{2}+4+4 x}{(x+4)^{2}}\right) d x$
$ =\int e^{x}\left(\frac{x}{x+4}+\frac{4}{(x+4)^{2}}\right) d x$
Let $g(x)=\frac{x}{(x+4)}$, then $g'(x)=\frac{4}{(x+4)^{2}}$
$=\int e^{x}\left\{g(x)+g^{\prime}(x)\right\} d x$
$ =e^{x} g(x)+c=e^{x}\left(\frac{x}{x+4}\right)+c$
$\therefore f(x)=\frac{x e^{x}}{x+4}$