What is the integral of (x^2e^x)/((2+x)^2) ?

The integral of (x^2e^x)/((2+x)^2) is -(e^xx^2)/(2+x)+e^xx-e^x+C

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integral of (x^2e^x)/((2+x)^2)

\int \frac{x ^{2} e ^{x}}{( 2 + x ) ^{2}} d x

- \frac{e ^{x} x ^{2}}{2 + x} + e^{x} x - e^{x} + C

Solution steps

\int \frac{x ^{2} e ^{x}}{( 2 + x ) ^{2}} d x

Apply Integration By Parts

= - \frac{e ^{x} x ^{2}}{2 + x} - \int - e^{x} x d x

\int - e^{x} x d x = - e^{x} x + e^{x}

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

Simplify

= - \frac{e ^{x} x ^{2}}{2 + x} + e^{x} x - e^{x}

Add a constant to the solution

= - \frac{e ^{x} x ^{2}}{2 + x} + e^{x} x - e^{x} + C

What is the integral of (x^2e^x)/((2+x)^2) ?
The integral of (x^2e^x)/((2+x)^2) is -(e^xx^2)/(2+x)+e^xx-e^x+C