What is the integral of-x^2e^{2x} ?

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

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integral of-x^2e^{2x}

\int - x^{2} e^{2 x} d x

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

Solution steps

\int - 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

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

Apply Integration By Parts

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

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

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

Simplify

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

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

Add a constant to the solution

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

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