Question

The value of the integral \[ \int e^x \left( \frac{1 - x}{1 + x^2} \right)^2 dx \] is

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

Hint:

For integrals involving rational functions and exponentials, use standard integration techniques or tables of integrals.

Answer 1

The Correct Option is C

Step 1: Integration process.
This integral simplifies after applying standard integration techniques involving exponential and rational functions. The result is \( \frac{e^x}{1 + x^2} + C \).

Step 2: Conclusion.
Thus, the correct answer is option (C).

Final Answer: \[ \boxed{\text{(C) } \frac{e^x}{1 + x^2} + C} \]