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

Tuesday, September 18, 2012

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,

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

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

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

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

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

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

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

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