What is the integral of x^2e^{ax} ?

The integral of x^2e^{ax} is e^{ax} 1/a x^2-2/a (e^{ax} 1/a x-e^{ax} 1/(a^2))+C

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integral of x^2e^{ax}

\int x^{2} e^{a x} d x

e^{a x} \frac{1}{a} x^{2} - \frac{2}{a} (e^{a x} \frac{1}{a} x - e^{a x} \frac{1}{a ^{2}}) + C

Solution steps

\int x^{2} e^{a x} d x

Apply Integration By Parts

: e^{a x} \frac{1}{a} x^{2} - \int e^{a x} \frac{2}{a} x d x

= e^{a x} \frac{1}{a} x^{2} - \int e^{a x} \frac{2}{a} x d x

\int e^{a x} \frac{2}{a} x d x = \frac{2}{a} (e^{a x} \frac{1}{a} x - e^{a x} \frac{1}{a ^{2}})

= e^{a x} \frac{1}{a} x^{2} - \frac{2}{a} (e^{a x} \frac{1}{a} x - e^{a x} \frac{1}{a ^{2}})

Add a constant to the solution

= e^{a x} \frac{1}{a} x^{2} - \frac{2}{a} (e^{a x} \frac{1}{a} x - e^{a x} \frac{1}{a ^{2}}) + C

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What is the integral of x^2e^{ax} ?
The integral of x^2e^{ax} is e^{ax} 1/a x^2-2/a (e^{ax} 1/a x-e^{ax} 1/(a^2))+C