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

The integral of (x+1)e^{4x} is 1/4 e^{4x}x+3/16 e^{4x}+C

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integral of (x+1)e^{4x}

\int (x + 1) e^{4 x} d x

\frac{1}{4} e^{4 x} x + \frac{3}{16} e^{4 x} + C

Solution steps

\int (x + 1) e^{4 x} d x

Apply Integration By Parts

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

\int \frac{1}{4} e^{4 x} d x = \frac{1}{16} e^{4 x}

= \frac{1}{4} e^{4 x} (x + 1) - \frac{1}{16} e^{4 x}

Simplify

\frac{1}{4} e^{4 x} (x + 1) - \frac{1}{16} e^{4 x} : \frac{1}{4} e^{4 x} x + \frac{3}{16} e^{4 x}

= \frac{1}{4} e^{4 x} x + \frac{3}{16} e^{4 x}

Add a constant to the solution

= \frac{1}{4} e^{4 x} x + \frac{3}{16} e^{4 x} + C

What is the integral of (x+1)e^{4x} ?
The integral of (x+1)e^{4x} is 1/4 e^{4x}x+3/16 e^{4x}+C