What is the integral of x^2e^{13x} ?

The integral of x^2e^{13x} is 1/13 e^{13x}x^2-2/2197 (13e^{13x}x-e^{13x})+C

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integral of x^2e^{13x}

\int x^{2} e^{13 x} d x

\frac{1}{13} e^{13 x} x^{2} - \frac{2}{2197} (13 e^{13 x} x - e^{13 x}) + C

Solution steps

\int x^{2} e^{13 x} d x

Apply Integration By Parts

= \frac{1}{13} e^{13 x} x^{2} - \int \frac{2}{13} e^{13 x} x d x

\int \frac{2}{13} e^{13 x} x d x = \frac{2}{2197} (13 e^{13 x} x - e^{13 x})

= \frac{1}{13} e^{13 x} x^{2} - \frac{2}{2197} (13 e^{13 x} x - e^{13 x})

Add a constant to the solution

= \frac{1}{13} e^{13 x} x^{2} - \frac{2}{2197} (13 e^{13 x} x - e^{13 x}) + C

What is the integral of x^2e^{13x} ?
The integral of x^2e^{13x} is 1/13 e^{13x}x^2-2/2197 (13e^{13x}x-e^{13x})+C