To find the integral \(\int x^2 e^{-x} \, dx\), we can use the method of integration by parts. Integration by parts is based on the formula:

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

We need to choose \(u\) and \(dv\) appropriately. Let's set:

\[ u = x^2 \quad \text{and} \quad dv = e^{-x} \, dx \]

Next, we need to find \(du\) and \(v\):

\[ du = 2x \, dx \quad \text{and} \quad v = \int e^{-x} \, dx = -e^{-x} \]

Now, apply the integration by parts formula:

\[ \int x^2 e^{-x} \, dx = uv - \int v \, du \]

Substitute \(u\), \(v\), \(du\), and \(dv\):

\[ \int x^2 e^{-x} \, dx = x^2 (-e^{-x}) - \int (-e^{-x}) (2x \, dx) \]
\[ = -x^2 e^{-x} + 2 \int x e^{-x} \, dx \]

Now we need to integrate \(\int x e^{-x} \, dx\) using integration by parts again. Set:

\[ u = x \quad \text{and} \quad dv = e^{-x} \, dx \]

Then:

\[ du = dx \quad \text{and} \quad v = -e^{-x} \]

Apply the integration by parts formula again:

\[ \int x e^{-x} \, dx = uv - \int v \, du \]

Substitute \(u\), \(v\), \(du\), and \(dv\):

\[ \int x e^{-x} \, dx = x (-e^{-x}) - \int (-e^{-x}) \, dx \]
\[ = -x e^{-x} + \int e^{-x} \, dx \]
\[ = -x e^{-x} - e^{-x} \]
\[ = -e^{-x} (x + 1) \]

Now substitute this result back into the original integral:

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

Therefore, the integral is:

\[ \int x^2 e^{-x} \, dx = -e^{-x} (x^2 + 2x + 2) + C \]

where \(C\) is the constant of integration.

Do you want more details or have any questions?

Here are some related questions for further practice:
1. Evaluate \(\int x e^{-x} \, dx\).
2. Evaluate \(\int x^3 e^{-x} \, dx\).
3. Find \(\int x^2 e^x \, dx\) using integration by parts.
4. Evaluate \(\int x^2 \cos(x) \, dx\).
5. Evaluate \(\int x^2 \sin(x) \, dx\).
6. Find \(\int x^2 e^{2x} \, dx\).
7. Evaluate \(\int x^2 \ln(x) \, dx\).
8. Find \(\int x^2 e^{-2x} \, dx\).

**Tip:** When using integration by parts, choose \(u\) and \(dv\) such that the resulting integral becomes simpler after applying the formula.

Learn how to calculate the integral \( \int x^2 e^{-x} \, dx \) using integration by parts. This detailed step-by-step solution covers the method of integration by parts, application of formulas, and simplification techniques. Suitable for advanced college-level students and anyone studying calculus.

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