Question

$\int \frac{1}{\sqrt{x}}\left(1+\frac{1}{x}\right)dx$

Answer 1

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