Question

The integral $ \int \sqrt{x^2 + a^2} \, dx $ is equal to

• A. \( \log |x + \sqrt{x^2 + a^2}| \)
• B. \( \frac{x \sqrt{x^2 + a^2}}{2} - \frac{a^2 \log |x + \sqrt{x^2 + a^2}|}{2} \)
• C. \( \frac{x \sqrt{x^2 + a^2}}{2} + \frac{a^2 \log |x + \sqrt{x^2 + a^2}|}{2} \)
• D. None of these

Hint:

For integrals involving \( \sqrt{x^2 + a^2} \), use the standard integral formula to simplify the process.

Answer 1

The Correct Option is C

Step 1: Recognizing the Integral Form
The given integral is: \[ \int \sqrt{x^2 + a^2} \, dx \] This is a standard integral, and the solution can be derived using the substitution method.
Step 2: Apply Integration Formula
The integral \( \int \sqrt{x^2 + a^2} \, dx \) has a standard solution: \[ \int \sqrt{x^2 + a^2} \, dx = \frac{x \sqrt{x^2 + a^2}}{2} + \frac{a^2 \log |x + \sqrt{x^2 + a^2}|}{2} \]
Step 3: Conclusion
Thus, the solution to the integral is \( \frac{x \sqrt{x^2 + a^2}}{2} + \frac{a^2 \log |x + \sqrt{x^2 + a^2}|}{2} \).