Question

Evaluate: \[ \int \sqrt{x^2 - a^2} \, dx. \]

Hint:

For integrals of the form \( \int \sqrt{x^2 - a^2} \, dx \), use the standard formula for such integrals involving square roots of quadratic expressions.

Answer 1

The integral \( \int \sqrt{x^2 - a^2} \, dx \) is a standard integral. To solve this, we use the following standard result: \[ \int \sqrt{x^2 - a^2} \, dx = \frac{x}{2} \sqrt{x^2 - a^2} - \frac{a^2}{2} \ln \left( x + \sqrt{x^2 - a^2} \right) + C, \] where \( C \) is the constant of integration. Conclusion: The value of the integral is: \[ \boxed{\frac{x}{2} \sqrt{x^2 - a^2} - \frac{a^2}{2} \ln \left( x + \sqrt{x^2 - a^2} \right) + C}. \]