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

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

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

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

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

Solution steps

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

Expand

(x^{2} + x) e^{2 x} : e^{2 x} x^{2} + e^{2 x} x

= \int e^{2 x} x^{2} + e^{2 x} x d x

Factor

e^{2 x} x^{2} + e^{2 x} x : e^{2 x} x (x + 1)

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

Apply Integration By Parts

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

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

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

Simplify

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

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

Add a constant to the solution

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

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