Question

The value of \( \int_0^\infty \frac{dx}{(a^2 + x^2)^7} \) is equal to

• A. \( \frac{231}{2047} \)
• B. \( \frac{1}{2048} \)
• C. \( \frac{232}{2047} \)
• D. \( \frac{231}{2048} \)

Hint:

Use standard integration techniques for rational functions of the form \( \frac{1}{(a^2 + x^2)^n} \).

Answer 1

The Correct Option is D

Step 1: Solving the integral.
The given integral can be solved using a standard formula for integrals of the form \( \int_0^\infty \frac{dx}{(a^2 + x^2)^n} \). Applying this, we get the value \( \frac{231}{2048} \).

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

Final Answer: \[ \boxed{\text{(D) } \frac{231}{2048}} \]