Solve: $\int_{2}^{3}\ \left(\sqrt {2x -\sqrt{5\left(4x-5\right)}}+\sqrt {2x +\sqrt{5\left(4x-5\right)}}\right)dx $

Question

How to solve following integration?

$$\int_{2}^{3}\ \left(\sqrt {2x -\sqrt{5\left(4x-5\right)}}+\sqrt {2x +\sqrt{5\left(4x-5\right)}}\right)dx $$

Answer 1

We pose $u=\sqrt{5(4x-5)}$ and we verify easly that the integral becomes $$\frac{\sqrt{10}}{50}\int_{\sqrt{15}}^{\sqrt{35}}u^2du.$$ I think you can easily complete the calculation.

Answer 2

Denote $y=\sqrt{5\left(4x-5\right)}$ and take the integrals of every summans separatively.