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

The integral of (3x+1)e^{2x} is 3/2 e^{2x}x-1/4 e^{2x}+C

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

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

\frac{3}{2} e^{2 x} x - \frac{1}{4} e^{2 x} + C

Solution steps

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

Apply Integration By Parts

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

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

\int \frac{3}{2} e^{2 x} d x = \frac{3}{4} e^{2 x}

= \frac{1}{2} e^{2 x} (3 x + 1) - \frac{3}{4} e^{2 x}

Simplify

\frac{1}{2} e^{2 x} (3 x + 1) - \frac{3}{4} e^{2 x} : \frac{3}{2} e^{2 x} x - \frac{1}{4} e^{2 x}

= \frac{3}{2} e^{2 x} x - \frac{1}{4} e^{2 x}

Add a constant to the solution

= \frac{3}{2} e^{2 x} x - \frac{1}{4} e^{2 x} + C

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What is the integral of (3x+1)e^{2x} ?
The integral of (3x+1)e^{2x} is 3/2 e^{2x}x-1/4 e^{2x}+C