Example 01 | The General Power Formula

$\displaystyle \int \left( \sqrt{x} + x\sqrt{x} + \dfrac{1}{\sqrt{x}} \right) \, dx$

$= \displaystyle \int \left( x^{1/2} + x \cdot x^{1/2} + \frac{1}{x^{1/2}} \right) \, dx$

$= \displaystyle \int (x^{1/2} + x^{3/2} + x^{-1/2}) \, dx$

$= \dfrac{x^{3/2}}{3/2} + \dfrac{x^{5/2}}{5/2} + \dfrac{x^{1/2}}{1/2} + C$

$= \frac{2}{3}x^{3/2} + \frac{2}{5}x^{5/2} + 2x^{1/2} + C$

$= \frac{2}{3}x\sqrt{x} + \frac{2}{5}x^2\sqrt{x} + 2\sqrt{x} + C$           answer