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

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

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integral of 4(x+2)e^{2x}

\int 4 (x + 2) e^{2 x} d x

2 e^{2 x} x + 3 e^{2 x} + C

Solution steps

\int 4 (x + 2) e^{2 x} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 4 \cdot \int (x + 2) e^{2 x} d x

Apply Integration By Parts

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

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

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

Simplify

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

= 2 e^{2 x} x + 3 e^{2 x}

Add a constant to the solution

= 2 e^{2 x} x + 3 e^{2 x} + C

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