Evaluate -2 5 x-4-5 x-2 d x - Vidyadip

Question

Evaluate $\int \frac{-2}{\sqrt{5 x-4}-\sqrt{5 x-2}} d x$

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Answer

$
\begin{aligned}
& \int \frac{-2}{\sqrt{5 x-4}-\sqrt{5 x-2}} d x \\
= & \int \frac{-2}{\sqrt{5 x-4}-\sqrt{5 x-2}} \times \frac{\sqrt{5 x-4}+\sqrt{5 x-2}}{\sqrt{5 x-4}+\sqrt{5 x-2}} d x \\
= & \int \frac{-2(\sqrt{5 x-4}+\sqrt{5 x-2})}{(5 x-4)-(5 x-2)} d x \\
= & \int(\sqrt{5 x-4}+\sqrt{5 x-2}) d x \\
= & \int(\sqrt{5 x-4})^{\frac{1}{2}} d x+\int(\sqrt{5 x-2})^{\frac{1}{2}} d x \\
= & \frac{(5 x-4)^{\frac{3}{2}}}{\left(\frac{3}{2}\right)} \times \frac{1}{5}+\frac{(5 x-2)^{\frac{3}{2}}}{\left(\frac{3}{2}\right)} \times \frac{1}{5}+c \\
= & \frac{2}{15}\left[(5 x-4)^{\frac{3}{2}}+(5 x-2)^{\frac{3}{2}}\right]+c .
\end{aligned}
$

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