Question

\[\int\frac{1}{\sqrt{3 x^2 + 5x + 7}} dx\]

Answer 1

\[\int\frac{dx}{\sqrt{3 x^2 + 5x + 7}}\]
\[ = \int\frac{dx}{\sqrt{3\left( x^2 + \frac{5}{3}x + \frac{7}{3} \right)}}\]
\[ = \frac{1}{\sqrt{3}}\int\frac{dx}{\sqrt{x^2 + \frac{5}{3}x + \left( \frac{5}{6} \right)^2 - \left( \frac{5}{6} \right)^2 + \frac{7}{3}}}\]
\[ = \frac{1}{\sqrt{3}}\int\frac{dx}{\sqrt{\left( x + \frac{5}{6} \right)^2 - \frac{25}{36} + \frac{7}{3}}}\]
\[ = \frac{1}{\sqrt{3}}\int\frac{dx}{\sqrt{\left( x + \frac{5}{6} \right)^2 + \frac{- 25 + 84}{36}}}\]

\[ = \frac{1}{\sqrt{3}}\int\frac{dx}{\sqrt{\left( x + \frac{5}{6} \right)^2 + \frac{59}{36}}}\]
\[ = \frac{1}{\sqrt{3}}\int\frac{dx}{\sqrt{\left( x + \frac{5}{6} \right)^2 + \left( \frac{\sqrt{59}}{36} \right)^2}}\]
\[ = \frac{1}{\sqrt{3}} \log \left| x + \frac{5}{6} + \sqrt{\left( x + \frac{5}{6} \right)^2 + \frac{59}{36}} \right| + C\]
\[ = \frac{1}{\sqrt{3}} \log \left| x + \frac{5}{6} + \sqrt{x^2 + \frac{5}{3}x + \frac{7}{3}} \right| + C\]