Question

$\int \frac{3}{\sqrt{7x-2}-\sqrt{7x-5}}dx$

Answer 1

We have,

$I={\int }^{}\frac{3}{\sqrt{7x-2}-\sqrt{7x-5}}dx$

$Rationalisethedenominator.=3\left[{\int }^{}\frac{\sqrt{7x-2}+\sqrt{7x-5}}{3}dx\right]=3\left[\frac{1}{3}{\int }^{}\sqrt{7x-2}dx+\frac{1}{3}{\int }^{}\sqrt{7x-5}dx\right]=3\left[\frac{2{\left(7x-2\right)}^{\frac{3}{2}}}{63}+\frac{2{\left(7x-5\right)}^{\frac{3}{2}}}{63}\right]=\frac{2\left({\left(7x-2\right)}^{\frac{3}{2}}+{\left(7x-5\right)}^{\frac{3}{2}}\right)}{21}+C$

Hence, this is the answer.