What is the integral of (2x+3)e^{5x} ?

The integral of (2x+3)e^{5x} is 2/5 e^{5x}x+13/25 e^{5x}+C

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integral of (2x+3)e^{5x}

\int (2 x + 3) e^{5 x} d x

\frac{2}{5} e^{5 x} x + \frac{13}{25} e^{5 x} + C

Solution steps

\int (2 x + 3) e^{5 x} d x

Apply Integration By Parts

: \frac{1}{5} e^{5 x} (2 x + 3) - \int \frac{2}{5} e^{5 x} d x

= \frac{1}{5} e^{5 x} (2 x + 3) - \int \frac{2}{5} e^{5 x} d x

\int \frac{2}{5} e^{5 x} d x = \frac{2}{25} e^{5 x}

= \frac{1}{5} e^{5 x} (2 x + 3) - \frac{2}{25} e^{5 x}

Simplify

\frac{1}{5} e^{5 x} (2 x + 3) - \frac{2}{25} e^{5 x} : \frac{2}{5} e^{5 x} x + \frac{13}{25} e^{5 x}

= \frac{2}{5} e^{5 x} x + \frac{13}{25} e^{5 x}

Add a constant to the solution

= \frac{2}{5} e^{5 x} x + \frac{13}{25} e^{5 x} + C

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What is the integral of (2x+3)e^{5x} ?
The integral of (2x+3)e^{5x} is 2/5 e^{5x}x+13/25 e^{5x}+C