Sure, let's go through an example of integration by parts.

### Example of Integration by Parts

To illustrate the method, let's consider the integral:

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

Integration by parts is based on the formula:

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

where \( u \) and \( dv \) are parts of the integrand chosen such that the resulting integral \( \int v \, du \) is simpler to evaluate.

#### Step-by-Step Solution:

1. **Choose \( u \) and \( dv \):**
   - Let \( u = x \), hence \( du = dx \).
   - Let \( dv = e^x \, dx \), hence \( v = e^x \).

2. **Apply the Integration by Parts Formula:**

   Substitute \( u \), \( du \), \( v \), and \( dv \) into the integration by parts formula:

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

3. **Evaluate the Remaining Integral:**

   The remaining integral is:

   \[
   \int e^x \, dx = e^x
   \]

4. **Combine the Results:**

   Substitute back into the integration by parts formula:

   \[
   \int x e^x \, dx = x e^x - e^x + C
   \]

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

#### Final Answer:

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

This is the result of the integral using the method of integration by parts.

Do you have any questions or need more details?

Here are five related questions you can ask:

1. Can you explain how to choose \( u \) and \( dv \) in integration by parts?
2. How do you solve an integral involving trigonometric functions using integration by parts?
3. What is the tabular method for repeated integration by parts?
4. How do you apply integration by parts to definite integrals?
5. Can you show an example of integrating a logarithmic function using integration by parts?

**Tip:** When using integration by parts, it is often helpful to choose \( u \) to be a function that becomes simpler when differentiated, and \( dv \) to be a function that is easy to integrate.

Explore an example of integration by parts with the integral \( \int x e^x \, dx \). Learn step-by-step how to apply the method, choosing appropriate \( u \) and \( dv \), and evaluating the resulting integral. Suitable for undergraduate students studying calculus.

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