What is the integral of (x^2+1)e^{3x} ?

The integral of (x^2+1)e^{3x} is 1/3 e^{3x}x^2-2/9 e^{3x}x+11/27 e^{3x}+C

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integral of (x^2+1)e^{3x}

\int (x^{2} + 1) e^{3 x} d x

\frac{1}{3} e^{3 x} x^{2} - \frac{2}{9} e^{3 x} x + \frac{11}{27} e^{3 x} + C

Solution steps

\int (x^{2} + 1) e^{3 x} d x

Apply Integration By Parts

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

Simplify

\frac{1}{3} e^{3 x} (x^{2} + 1) - \frac{2}{27} (3 e^{3 x} x - e^{3 x}) : \frac{1}{3} e^{3 x} x^{2} - \frac{2}{9} e^{3 x} x + \frac{11}{27} e^{3 x}

= \frac{1}{3} e^{3 x} x^{2} - \frac{2}{9} e^{3 x} x + \frac{11}{27} e^{3 x}

Add a constant to the solution

= \frac{1}{3} e^{3 x} x^{2} - \frac{2}{9} e^{3 x} x + \frac{11}{27} e^{3 x} + C

What is the integral of (x^2+1)e^{3x} ?
The integral of (x^2+1)e^{3x} is 1/3 e^{3x}x^2-2/9 e^{3x}x+11/27 e^{3x}+C