Question

Write the primitive or anti-derivative of $f\left(x\right)=\sqrt{x}+\frac{1}{\sqrt{x}}.$

Answer 1

$f\left(x\right)=\sqrt{x}+\frac{1}{\sqrt{x}}$
Integrating both sides
$$\begin{gathered} \int f\left(x\right)dx=\int \left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)dx \\ =\int \left(x^{\frac{1}{2}}+x^{-\frac{1}{2}}\right)dx \\ =\left[\frac{x^{{\displaystyle \frac{1}{2}}+1}}{{\displaystyle \frac{1}{2}}+1}\right]+\left[\frac{x^{-{\displaystyle \frac{1}{2}}+1}}{-{\displaystyle \frac{1}{2}}+1}\right]+\mathrm{C} \\ =\frac{2}{3}{x}^{\frac{3}{2}}+2x^{\frac{1}{2}}+\mathrm{C} \end{gathered}$$