Question

Write the anti-derivative of $\left(3\sqrt{x}+\frac{1}{\sqrt{x}}\right).$

Answer 1

$$\begin{gathered} \int \left(3\sqrt{x}+\frac{1}{\sqrt{x}}\right)\mathrm{d}x=3\frac{x^{{\displaystyle \frac{1}{2}}+1}}{{\displaystyle \frac{1}{2}}+1}+\frac{x^{-{\displaystyle \frac{1}{2}}+1}}{-{\displaystyle \frac{1}{2}}+1}+c \\ =2x^{\frac{3}{2}}+2x^{\frac{1}{2}}+c \\ =2\left(x^{\frac{3}{2}}+x^{\frac{1}{2}}\right)+c \\ \mathrm{Hence},\mathrm{the}\mathrm{anti}-\mathrm{derivative}\mathrm{of}\left(3\sqrt{\mathrm{x}}+\frac{1}{\sqrt{\mathrm{x}}}\right)\mathrm{is}2\left(x^{\frac{3}{2}}+x^{\frac{1}{2}}\right)+c. \end{gathered}$$