Question

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

Answer 1

\[\int\frac{1}{\sqrt{x}}\left( 1 + \frac{1}{x} \right)dx\]
\[ = \int x^{- \frac{1}{2}} \left( 1 + \frac{1}{x} \right)dx\]
\[ = \int\left( x^{- \frac{1}{2}} + \frac{1}{x^\frac{3}{2}} \right)dx\]
\[ = \int x^{- \frac{1}{2}} dx + \int x^{- \frac{3}{2}} dx\]
\[ = \left[ \frac{x^{- \frac{1}{2} + 1}}{- \frac{1}{2} + 1} \right] + \left[ \frac{x^{- \frac{3}{2} + 1}}{- \frac{3}{2} + 1} \right]\]
\[ = 2\sqrt{x} - \frac{2}{\sqrt{x}} + C\]