1 2 x^2-5 d x - Vidyadip

Question

$\int \frac{1}{\sqrt{2 x^2-5}} d x$

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Answer

$\text { Let } I =\int \frac{1}{\sqrt{2 x^2-5}} d x$
$=\int \frac{1}{\sqrt{2\left(x^2-\frac{5}{2}\right)}} d x$
$=\frac{1}{\sqrt{2}} \int \frac{1}{\sqrt{x^2-\left(\frac{\sqrt{5}}{\sqrt{2}}\right)^2}} d x$
$=\frac{1}{\sqrt{2}} \log \left|x+\sqrt{x^2-\left(\frac{\sqrt{5}}{\sqrt{2}}\right)^2 \mid+ c }\right|$
$\therefore I =\frac{1}{\sqrt{2}} \log \left|x+\sqrt{x^2-\frac{5}{2}}\right|+ c $

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