Question

The value of \( \int_{0}^{1} x e^{x} \, dx \) is:

• A. 1
• B. $e-1$
• C. $e$
• D. 0

Hint:

The integral $\int x e^x dx$ is a common result used to derive many other exponential integrals.

Answer 1

The Correct Option is A

Step 1: Concept
Use Integration by Parts: $\int u v dx = u \int v dx - \int (u' \int v dx) dx$.
Step 2: Analysis
Let $u = x, v = e^x$.
$\int x e^x dx = x e^x - \int 1 \cdot e^x dx = x e^x - e^x$.
Applying limits: $[(1 \cdot e^1 - e^1) - (0 \cdot e^0 - e^0)]$.
Step 3: Conclusion
$= (e - e) - (0 - 1) = 0 + 1 = 1$.
Final Answer: (A)