Question

The integral $ \int \frac{x e^x}{(1+x)^2} dx $ is equal to

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

Hint:

When encountering complex integrals, use integration by parts and simplify the terms carefully to reach the final solution.

Answer 1

The Correct Option is B

Step 1: Apply Integration by Parts Let: \[ u = \frac{1}{(1+x)^2}, \quad dv = x e^x dx \] Using the integration by parts formula: \[ \int u dv = uv - \int v du \]
Step 2: Simplify the Integral After performing the necessary integrations and simplifications, we arrive at the final expression for the integral: \[ \int \frac{x e^x}{(1+x)^2} dx = \frac{1 + 2x e^x}{1+x} \]
Step 3: Conclusion Thus, the integral is \( \frac{1 + 2x e^x}{1+x} \).