Question

Evaluate the integral \( \int x e^{x^2} dx \):

• A. \( \frac{1}{2} e^{x^2} + C \)
• B. \( e^{x^2} + C \)
• C. \( \frac{1}{2} x e^{x^2} + C \)
• D. \( x^2 e^{x^2} + C \)

Hint:

Key Fact: Look for substitution opportunities when the integrand includes a function and its derivative.

Answer 1

The Correct Option is A

To evaluate the integral \( \int x e^{x^2} \, dx \), we will use the method of substitution. Let's set \( u = x^2 \). Then, the derivative of \( u \) with respect to \( x \) is \( \frac{du}{dx} = 2x \), which implies that \( du = 2x \, dx \) or \( \frac{1}{2} du = x \, dx \).

Substituting these into the integral, we have:

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

This simplifies to:

\[ \frac{1}{2} \int e^u \, du \]

The integral of \( e^u \) with respect to \( u \) is simply \( e^u + C \). Thus, the expression becomes:

\[ \frac{1}{2} (e^u + C) = \frac{1}{2} e^u + C \]

Since \( u = x^2 \), we substitute back to get:

\[ \frac{1}{2} e^{x^2} + C \]

This leads us to the correct solution: \( \frac{1}{2} e^{x^2} + C \)

Answer 2

Step 1: Use substitution method
Let \( u = x^2 \Rightarrow du = 2x dx \Rightarrow x dx = \frac{1}{2} du \)

Step 2: Substitute and integrate 
\[ \int x e^{x^2} dx = \int e^u \cdot \frac{1}{2} du = \frac{1}{2} \int e^u du = \frac{1}{2} e^u + C \] \[ \Rightarrow \frac{1}{2} e^{x^2} + C \]