Question

Evaluate the integral \[ \int x^3 e^{x^2} \, dx \]

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

Hint:

For integrals of the form \( x^3 e^{x^2} \), use substitution \( u = x^2 \) and integrate accordingly.

Answer 1

The Correct Option is B

Step 1: Use substitution.
Let \( u = x^2 \), so that \( du = 2x \, dx \). The integral becomes: \[ \int x^3 e^{x^2} \, dx = \frac{1}{2} \int e^u (u + 1) \, du \]
Step 2: Solve the integral.
Integrating, we get: \[ \frac{1}{2} e^{x^2} (x^2 - 1) + c \]
Step 3: Conclusion.
Thus, the solution to the integral is \( \frac{1}{2} e^{x^2} (x^2 - 1) + c \), corresponding to option (B).