Question

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

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

Hint:

For integrals like \( \int x e^{x^2} \, dx \), try using substitution to simplify the integrand.

Answer 1

The Correct Option is D

To solve \( \int x e^{x^2} \, dx \), we use the substitution method. 
Let: \[ u = x^2 \quad {so} \quad du = 2x \, dx \] Thus: \[ \frac{du}{2} = x \, dx \] The integral becomes: \[ \int x e^{x^2} \, dx = \frac{1}{2} \int e^u \, du = \frac{e^{u}}{2} = \frac{e^{x^2}}{2} + C \]