To evaluate the integral \( R = \int_{0}^{2} x e^{x^2} \, dx \), we can use the technique of substitution.

### Step 1: Substitution
Let \( u = x^2 \). Then, the derivative of \( u \) with respect to \( x \) is:
\[
\frac{du}{dx} = 2x
\]
So, \( du = 2x \, dx \), or \( \frac{1}{2} du = x \, dx \).

### Step 2: Rewrite the Integral
Substitute into the integral:
\[
R = \int_{0}^{2} x e^{x^2} \, dx = \int_{u(0)}^{u(2)} e^{u} \cdot \frac{1}{2} du
\]
Given \( u = x^2 \), when \( x = 0 \), \( u = 0 \), and when \( x = 2 \), \( u = 4 \). Therefore, the integral becomes:
\[
R = \frac{1}{2} \int_{0}^{4} e^u \, du
\]

### Step 3: Evaluate the Integral
The integral of \( e^u \) with respect to \( u \) is simply \( e^u \). So, we have:
\[
R = \frac{1}{2} \left[ e^u \right]_{0}^{4} = \frac{1}{2} \left( e^4 - e^0 \right)
\]

### Step 4: Simplify the Result
Since \( e^0 = 1 \), the final expression is:
\[
R = \frac{1}{2} \left( e^4 - 1 \right)
\]

So, the evaluated integral is:
\[
R = \frac{e^4 - 1}{2}
\]

Would you like more details or have any questions?

### Related Questions:
1. How would the integral change if the limits were different, say from 1 to 3?
2. Can the method of integration by parts be applied to this integral?
3. How does substitution work in integrals involving trigonometric functions?
4. What is the significance of the constant factor in definite integrals?
5. How does the exponential function behave under integration compared to polynomial functions?

**Tip:** Substitution is a powerful technique for solving integrals where the integrand is a composite function, especially when one part of the function is the derivative of another.

Learn how to evaluate the integral ∫(0 to 2) x e^(x^2) dx step-by-step using the substitution method. This problem involves substitution in integrals and explores the behavior of exponential functions under integration. Suitable for advanced high school and college students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation