Question

Evaluate the integral: \[ I = \int_0^x \frac{t^2}{\sqrt{a^2 + t^2}} dt \]

• A. \( \frac{x}{2} \sqrt{a^2 + x^2} + \log \left| x + \sqrt{a^2 + x^2} \right| \)
• B. \( \sqrt{a^2 + x^2} - a^2 \operatorname{Sinh}^{-1} \frac{x}{a} \)
• C. \( \frac{x}{2} \sqrt{a^2 + x^2} - \frac{a^2}{4} \log \left| x + \sqrt{a^2 + x^2} \right| \)
• D. \( \frac{x}{2} \sqrt{a^2 + x^2} - \frac{a^2}{2} \operatorname{Sinh}^{-1} \frac{x}{a} \)

Hint:

For integrals involving square roots of quadratic expressions, use trigonometric or hyperbolic substitutions like: \[ t = a \sinh u, t = a \tan \theta \] for simplification.

Answer 1

The Correct Option is D

Using the substitution \( t = a \sinh u \), we rewrite: \[ dt = a \cosh u \, du \] Substituting and simplifying, integrating step by step: \[ I = \int_0^x \frac{t^2}{\sqrt{a^2 + t^2}} dt = \frac{x}{2} \sqrt{a^2 + x^2} - \frac{a^2}{2} \operatorname{Sinh}^{-1} \frac{x}{a} \] Thus, the correct answer is: \[ \frac{x}{2} \sqrt{a^2 + x^2} - \frac{a^2}{2} \operatorname{Sinh}^{-1} \frac{x}{a} \]