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

The integral of (x-x^2)e^{3x} is -1/3 e^{3x}x^2+5/9 e^{3x}x-5/27 e^{3x}+C

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

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

- \frac{1}{3} e^{3 x} x^{2} + \frac{5}{9} e^{3 x} x - \frac{5}{27} e^{3 x} + C

Solution steps

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

Expand

(x - x^{2}) e^{3 x} : e^{3 x} x - e^{3 x} x^{2}

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int e^{3 x} x d x = \frac{1}{9} (e^{3 x} \cdot 3 x - e^{3 x})

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

= \frac{1}{9} (e^{3 x} \cdot 3 x - e^{3 x}) - (\frac{1}{3} e^{3 x} x^{2} - \frac{2}{27} (3 e^{3 x} x - e^{3 x}))

Simplify

\frac{1}{9} (e^{3 x} \cdot 3 x - e^{3 x}) - (\frac{1}{3} e^{3 x} x^{2} - \frac{2}{27} (3 e^{3 x} x - e^{3 x})) : - \frac{1}{3} e^{3 x} x^{2} + \frac{5}{9} e^{3 x} x - \frac{5}{27} e^{3 x}

= - \frac{1}{3} e^{3 x} x^{2} + \frac{5}{9} e^{3 x} x - \frac{5}{27} e^{3 x}

Add a constant to the solution

= - \frac{1}{3} e^{3 x} x^{2} + \frac{5}{9} e^{3 x} x - \frac{5}{27} e^{3 x} + C

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What is the integral of (x-x^2)e^{3x} ?
The integral of (x-x^2)e^{3x} is -1/3 e^{3x}x^2+5/9 e^{3x}x-5/27 e^{3x}+C