Question

Write a value of\[\int\sqrt{9 + x^2} \text{ dx }\].

Answer 1

\[\int \sqrt{9 + x^2} \text{ dx }\]
\[ = \int \sqrt{3^2 + x^2} dx \left( \because \sqrt{a^2 + x^2} = \frac{x}{2}\sqrt{x^2 + a^2} + \frac{a^2}{2}\text{ ln }\left| x + \sqrt{x^2 + a^2} \right| \right)\]
\[ = \frac{x}{2}\sqrt{9 + x^2} + \frac{9}{2}\text{ ln }\left| x + \sqrt{9 + x^2} \right| + C\]