Question

\[\int\frac{1}{\sqrt{a^2 + b^2 x^2}} dx\]

Answer 1

\[\int\frac{dx}{\sqrt{a^2 + b^2 x^2}}\]
\[ = \int\frac{dx}{\sqrt{b^2 \left( \frac{a^2}{b^2} + x^2 \right)}}\]
\[ = \frac{1}{b}\int\frac{dx}{\sqrt{x^2 + \left( \frac{a}{b} \right)^2}}\]
\[ = \frac{1}{b} \text{  log }\left| x + \sqrt{x^2 + \frac{a^2}{b^2}} \right| + C\]
\[ = \frac{1}{b}\left[ \text{ log }\left| x + \frac{\sqrt{b^2 x^2 + a^2}}{b} \right| \right] + C\]
\[ = \frac{1}{b}\left[ \text{  log }\left| \frac{bx + \sqrt{b^2 x^2 + a^2}}{b} \right| \right] + C\]
\[ = \frac{1}{b}\left[ \text{ log }\left| bx + \sqrt{b^2 x^2 + a^2} \right| - \text{ log b }\right] + C\]
\[ = \frac{1}{b} \text{ log }\left| bx + \sqrt{b^2 x^2 + a^2} \right| - \frac{\log b}{b} + C\]
\[\text{  let C} - \frac{\log b}{b} = C'\]
\[ = \frac{1}{b}\text{ log }\left| bx + \sqrt{b^2 x^2 + a^2} \right| + C'\]