Question

\[\int\sqrt{16 x^2 + 25} \text{ dx}\]

Answer 1

\[\int \sqrt{16 x^2 + 25} \text{ dx}\]
\[ = \int \sqrt{16\left( x^2 + \frac{25}{16} \right)}\text{ dx}\]
\[ = 4\int \sqrt{x^2 + \left( \frac{5}{4} \right)^2} \text{ dx}\]
\[ = 4\left[ \frac{x}{2}\sqrt{x^2 + \left( \frac{5}{4} \right)^2} + \frac{\left( \frac{5}{4} \right)^2}{2}\text{ ln }\left| x + \sqrt{x^2 + \left( \frac{5}{4} \right)^2} \right| \right] + C \left[ \because \int\sqrt{x^2 + a^2} \text{ dx} = \frac{1}{2}x\sqrt{x^2 + a^2} + \frac{1}{2} a^2 \text{ ln}\left| x + \sqrt{x^2 + a^2} \right| + C \right]\]
\[ = 2x \sqrt{x^2 + \frac{25}{16}} + \frac{25}{8}\text{ ln }\left| x + \sqrt{x^2 + \frac{25}{16}} \right| + C\]