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

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