# How do you integrate #x^2 e^-5x dx#?

Hello,

Utilize the formula for integration by parts:

You could make things simpler:

Lastly, enter that into the expression above:

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To integrate ( x^2 e^{-5x} , dx ), you can use integration by parts. Let ( u = x^2 ) and ( dv = e^{-5x} , dx ). Then, ( du = 2x , dx ) and ( v = -\frac{1}{5} e^{-5x} ).

Now, applying the integration by parts formula:

[ \int u , dv = uv - \int v , du ]

We have:

[ \int x^2 e^{-5x} , dx = -\frac{x^2}{5} e^{-5x} - \int -\frac{2x}{5} e^{-5x} , dx ]

Integrating the second term again by parts:

Let ( u = x ) and ( dv = e^{-5x} , dx ). Then, ( du = dx ) and ( v = -\frac{1}{5} e^{-5x} ).

[ -\frac{x^2}{5} e^{-5x} - \left( -\frac{x}{5} e^{-5x} - \int -\frac{1}{5} e^{-5x} , dx \right) ]

[ -\frac{x^2}{5} e^{-5x} + \frac{x}{5} e^{-5x} - \frac{1}{25} e^{-5x} + C ]

So, the integral of ( x^2 e^{-5x} , dx ) is ( -\frac{x^2}{5} e^{-5x} + \frac{x}{5} e^{-5x} - \frac{1}{25} e^{-5x} + C ), where ( C ) is the constant of integration.

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