Question

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

Answer 1

$$\begin{gathered} \int \frac{dx}{\sqrt{3x^2+5x+7}} \\ =\int \frac{dx}{\sqrt{3\left(x^2+{\displaystyle \frac{5}{3}}x+{\displaystyle \frac{7}{3}}\right)}} \\ =\frac{1}{\sqrt{3}}\int \frac{dx}{\sqrt{x^2+{\displaystyle \frac{5}{3}}x+{\left({\displaystyle \frac{5}{6}}\right)}^2-{\left({\displaystyle \frac{5}{6}}\right)}^2+{\displaystyle \frac{7}{3}}}} \\ =\frac{1}{\sqrt{3}}\int \frac{dx}{\sqrt{{\left(x+{\displaystyle \frac{5}{6}}\right)}^2-{\displaystyle \frac{25}{36}}+{\displaystyle \frac{7}{3}}}} \\ =\frac{1}{\sqrt{3}}\int \frac{dx}{\sqrt{{\left(x+{\displaystyle \frac{5}{6}}\right)}^2+{\displaystyle \frac{-25+84}{36}}}} \\ =\frac{1}{\sqrt{3}}\int \frac{dx}{\sqrt{{\left(x+{\displaystyle \frac{5}{6}}\right)}^2+{\displaystyle \frac{59}{36}}}} \\ =\frac{1}{\sqrt{3}}\int \frac{dx}{\sqrt{{\left(x+{\displaystyle \frac{5}{6}}\right)}^2+{\left({\displaystyle \frac{\sqrt{59}}{36}}\right)}^2}} \\ =\frac{1}{\sqrt{3}}\log \left|x+\frac{5}{6}+\sqrt{{\left(x+\frac{5}{6}\right)}^2+\frac{59}{36}}\right|+\mathrm{C} \\ =\frac{1}{\sqrt{3}}\log \left|x+\frac{5}{6}+\sqrt{x^2+\frac{5}{3}x+\frac{7}{3}}\right|+\mathrm{C} \end{gathered}$$