∫(x^2e^x)/((x+2)^2) dx is equal to

The correct answer is Option (2) → \(\displaystyle e^x\left(\frac{x-2}{x+2}\right)+C\)

Writing \(x^2 = (x+2)^2-4(x+2)+4\) splits the fraction:

    \( \displaystyle \frac{x^2}{(x+2)^2} = 1-\frac{4}{x+2}+\frac{4}{(x+2)^2} \)

    \( \displaystyle = \frac{x-2}{x+2}+\frac{4}{(x+2)^2} \)

With \(f(x) = \frac{x-2}{x+2} = 1-\frac{4}{x+2}\), the derivative is

    \( \displaystyle f'(x) = \frac{4}{(x+2)^2} \)

So the integrand is \(e^x\left(f(x)+f'(x)\right)\), and the e^x(f + f’) result \(\int e^x\left(f+f'\right)dx = e^xf(x)+C\) gives

    \( \displaystyle \int\frac{x^2e^x}{(x+2)^2}\,dx = e^x\left(\frac{x-2}{x+2}\right)+C \)

Why the other options are incorrect

• Option (1) \(\displaystyle e^x\left(\frac{x+2}{x-2}\right)+C\) — Incorrect. The fraction is upside down; its derivative has \((x-2)^2\) in the denominator.

• Option (3) \(\displaystyle e^x\left(\frac{x}{x+2}\right)+C\) — Incorrect. Its derivative is \(\frac{e^x\left(x^2+2x+2\right)}{(x+2)^2}\).

• Option (4) \(\displaystyle e^x\left(\frac{x^2}{x+2}\right)+C\) — Incorrect. Its derivative is \(\frac{e^x\left(x^3+3x^2+4x\right)}{(x+2)^2}\).