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 \( \int x e^{x^2} \, dx \), employ substitution. Set \( u = x^2 \). Differentiating \( u \) with respect to \( x \) yields \( \frac{du}{dx} = 2x \), so \( du = 2x \, dx \), or \( \frac{1}{2} du = x \, dx \).

Substituting into the integral results in:

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

This simplifies to:

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

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

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

Substitute back \( u = x^2 \) to obtain:

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

The final solution is \( \frac{1}{2} e^{x^2} + C \).