Evaluate: 3 7 x-2-7 x-5 d x - Vidyadip

Question

Evaluate:
$\int \frac{3}{\sqrt{7 x-2}-\sqrt{7 x-5}} \cdot d x$

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Answer

$\int \frac{3}{\sqrt{7 x-2}-\sqrt{7 x-5}} d x$
$=\int \frac{3}{\sqrt{7 x-2}-\sqrt{7 x-5}} \times \frac{\sqrt{7 x-2}+\sqrt{7 x-5}}{\sqrt{7 x-2}-\sqrt{7 x-5}} d x$
$=\int \frac{3(\sqrt{7 x-2}+\sqrt{7 x-5})}{(7 x-2)-(7 x-5)} d x$
$=\int(\sqrt{7 x-2}+\sqrt{7 x-5}) d x$
$=\int(7 x-2)^{\frac{1}{2}} d x+\int(7 x-5)^{\frac{1}{2}} d x$
$=\frac{(7 x-2)^{\frac{3}{2}}}{3 / 2} \times \frac{1}{7}+\frac{(7 x-5)^{\frac{3}{2}}}{3 / 2} \times \frac{1}{7}+c$
$=\frac{2}{21}(7 x-2)^{\frac{3}{2}}+\frac{2}{21}(7 x-5)^{\frac{2}{2}}+c .$

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