What is the integral of (x^2e^{3x}) ?

The integral of (x^2e^{3x}) is 1/3 e^{3x}x^2-2/27 (3e^{3x}x-e^{3x})+C

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integral of (x^2e^{3x})

\int (x^{2} e^{3 x}) d x

\frac{1}{3} e^{3 x} x^{2} - \frac{2}{27} (3 e^{3 x} x - e^{3 x}) + C

Solution steps

\int x^{2} e^{3 x} d x

Apply Integration By Parts

= \frac{1}{3} e^{3 x} x^{2} - \int \frac{2}{3} e^{3 x} x d x

\int \frac{2}{3} e^{3 x} x d x = \frac{2}{27} (3 e^{3 x} x - e^{3 x})

= \frac{1}{3} e^{3 x} x^{2} - \frac{2}{27} (3 e^{3 x} x - e^{3 x})

Add a constant to the solution

= \frac{1}{3} e^{3 x} x^{2} - \frac{2}{27} (3 e^{3 x} x - e^{3 x}) + C

What is the integral of (x^2e^{3x}) ?
The integral of (x^2e^{3x}) is 1/3 e^{3x}x^2-2/27 (3e^{3x}x-e^{3x})+C