Question

$\int e^x{\left(\frac{x+2}{x+4}\right)}^{2}dx$ is equal to

• A. $\frac{x{e}^x}{x+4}+c$
• B. $e^x\left(\frac{x+2}{x+4}\right)+c$
• C. $\frac{e^x}{x+4}+c$
• D. none of these

Answer 1

The correct option is A $\frac{x{e}^x}{x+4}+c$

$\int e^x{\left(\frac{x+2}{x+4}\right)}^{2}dx=\int e^x\left(\frac{x^2+4x+4}{{(x+4)}^2}\right)dx$

$=\int e^x(\frac{x}{x+4}+\frac{4}{{(x+4)}^2})dx$

$=\int e^x[\frac{x}{x+4}+\frac{d}{dx}\left(\frac{x}{x+4}\right)]dx$

$=\frac{x{e}^x}{x+4}+c$