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We are asked to evaluate the integral:

\[
\int -3x e^x \, dx
\]

This integral can be solved using **integration by parts**, which follows the formula:

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

Here, we need to choose parts for \(u\) and \(dv\). Let's make the following choices:

- \( u = x \), which implies \( du = dx \)
- \( dv = e^x \, dx \), which implies \( v = e^x \)

Now apply the integration by parts formula:

\[
\int -3x e^x \, dx = -3 \left( x e^x - \int e^x \, dx \right)
\]

We know that \( \int e^x \, dx = e^x \), so:

\[
= -3 \left( x e^x - e^x \right) + C
\]

Simplifying:

\[
= -3 \left( e^x (x - 1) \right) + C
\]

Thus, the solution to the integral is:

\[
-3e^x (x - 1) + C
\]

Would you like more details or have any questions?

Here are five related questions:
1. How does integration by parts work in general?
2. Can you solve an integral using substitution instead of parts?
3. How does the choice of \(u\) and \(dv\) affect the outcome in integration by parts?
4. What happens if we apply integration by parts twice in a complex integral?
5. How would you integrate \(x^2 e^x\)?

**Tip:** When applying integration by parts, always choose \(u\) to be a function that simplifies upon differentiation.

Evaluate the integral \( \int -3x e^x \, dx \)

Learn how to solve the integral \( \int -3x e^x \, dx \) using integration by parts. This detailed solution breaks down the steps involved, including choosing the appropriate functions for \(u\) and \(dv\), applying the formula, and simplifying the final expression.

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