`int 2 / (xsqrt(9x^2-25)) dx` Find the indefinite integral

Given the integral: `\int \frac{2}{x\sqrt{9x^2-25}}dx`

Let `x=\frac{5}{3}sect` ``

So, `dx=\frac{5}{3}sect tant dt`

Hence we have,

`\int \frac{2}{x\sqrt{9x^2-25}}dx=\int \frac{\frac{10}{3}sect tant}{\frac{5}{3}sec t\sqrt{25sec^2t-25}}dt`

                      `=\int \frac{2tant}{\sqrt{25(sec^2t-1)}}dt`

                       `=\int \frac{2tan t}{\sqrt{25tan^2t}}dt`

                       `=\int \frac{2tant}{5tant}dt`

                       `=\frac{2}{5}\int dt`

                        `=\frac{2}{5}t+C`

                         `=\frac{2}{5}sec^{-1}(\frac{3}{5}x)+C`