https://file.aiquickdraw.com/mathbot/file/1729136809112hbvshz88.jpg

To solve the integral \( \int x^2 e^{2x} \, dx \) using integration by parts, let's break down the steps methodically. Integration by parts follows the formula:

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

### Step 1: Choose \( u \) and \( dv \)
We want to simplify the integral by reducing the power of \( x^2 \), so let's choose:
- \( u = x^2 \), which implies \( du = 2x \, dx \)
- \( dv = e^{2x} \, dx \), which implies \( v = \frac{1}{2} e^{2x} \) (since \( \int e^{2x} \, dx = \frac{1}{2} e^{2x} \))

### Step 2: Apply the integration by parts formula
Now apply the formula:

\[
\int x^2 e^{2x} \, dx = x^2 \cdot \frac{1}{2} e^{2x} - \int \frac{1}{2} e^{2x} \cdot 2x \, dx
\]

Simplifying:

\[
= \frac{1}{2} x^2 e^{2x} - \int x e^{2x} \, dx
\]

### Step 3: Apply integration by parts again
Now we need to solve \( \int x e^{2x} \, dx \) using integration by parts again. Choose:
- \( u = x \), so \( du = dx \)
- \( dv = e^{2x} \, dx \), so \( v = \frac{1}{2} e^{2x} \)

Apply the formula:

\[
\int x e^{2x} \, dx = x \cdot \frac{1}{2} e^{2x} - \int \frac{1}{2} e^{2x} \, dx
\]

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

### Step 4: Substitute back
Now substitute this result into the original equation:

\[
\int x^2 e^{2x} \, dx = \frac{1}{2} x^2 e^{2x} - \left( \frac{1}{2} x e^{2x} - \frac{1}{4} e^{2x} \right)
\]

Simplify the expression:

\[
\int x^2 e^{2x} \, dx = \frac{1}{2} x^2 e^{2x} - \frac{1}{2} x e^{2x} + \frac{1}{4} e^{2x} + C
\]

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

### Final Answer:
\[
\int x^2 e^{2x} \, dx = \frac{1}{2} x^2 e^{2x} - \frac{1}{2} x e^{2x} + \frac{1}{4} e^{2x} + C
\]

Do you have any questions or need further clarification on any of the steps?

Here are five questions for further exploration:
1. How do we determine the best choices for \( u \) and \( dv \) in integration by parts?
2. What happens if we choose different values for \( u \) and \( dv \) in this problem?
3. Can we verify the solution by differentiating the final result?
4. What is the difference between using integration by parts once versus multiple times in a problem like this?
5. How does the constant of integration \( C \) affect the final answer in indefinite integrals?

**Tip**: When solving integrals involving both polynomials and exponentials, integration by parts is typically an efficient method to reduce the polynomial term step by step.

Learn how to evaluate the integral ∫ x² e^{2x} dx using the method of integration by parts. This solution includes a detailed step-by-step breakdown and is suitable for college-level calculus students. The problem covers key calculus concepts such as integration by parts and exponential functions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation