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

The integral of e^{3x}(2x+3) is 2/3 e^{3x}x+7/9 e^{3x}+C

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

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

\frac{2}{3} e^{3 x} x + \frac{7}{9} e^{3 x} + C

Solution steps

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

Apply Integration By Parts

: \frac{2}{3} e^{3 x} x + e^{3 x} - \int \frac{2}{3} e^{3 x} d x

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

\int \frac{2}{3} e^{3 x} d x = \frac{2}{9} e^{3 x}

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

Add similar elements:

e^{3 x} - \frac{2}{9} e^{3 x} = \frac{7}{9} e^{3 x}

= \frac{2}{3} e^{3 x} x + \frac{7}{9} e^{3 x}

Add a constant to the solution

= \frac{2}{3} e^{3 x} x + \frac{7}{9} e^{3 x} + C

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What is the integral of e^{3x}(2x+3) ?
The integral of e^{3x}(2x+3) is 2/3 e^{3x}x+7/9 e^{3x}+C