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

The integral of (x+1)e^{2x+1} is 1/2 e^{2x+1}x+1/4 e^{2x+1}+C

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

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

\frac{1}{2} e^{2 x + 1} x + \frac{1}{4} e^{2 x + 1} + C

Solution steps

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

Apply Integration By Parts

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

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

\int \frac{1}{2} e^{2 x + 1} d x = \frac{1}{4} e^{2 x + 1}

= \frac{1}{2} e^{2 x + 1} (x + 1) - \frac{1}{4} e^{2 x + 1}

Simplify

\frac{1}{2} e^{2 x + 1} (x + 1) - \frac{1}{4} e^{2 x + 1} : \frac{1}{2} e^{2 x + 1} x + \frac{1}{4} e^{2 x + 1}

= \frac{1}{2} e^{2 x + 1} x + \frac{1}{4} e^{2 x + 1}

Add a constant to the solution

= \frac{1}{2} e^{2 x + 1} x + \frac{1}{4} e^{2 x + 1} + C

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What is the integral of (x+1)e^{2x+1} ?
The integral of (x+1)e^{2x+1} is 1/2 e^{2x+1}x+1/4 e^{2x+1}+C