Answer
$ \frac{2}{3}x^{\frac{3}{2}} + x^{\frac{1}{2}} + C $
Work Step by Step
$\int (\sqrt x + \frac{1}{2\sqrt x})dx$
= $\int \sqrt x dx + \int\frac{1}{2\sqrt x}dx$
= $\int x^{\frac{1}{2}} dx + \int\frac{1}{2}\frac{1}{\sqrt x}dx$
=$ \frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1} + C' + \frac{1}{2}\int x^{-\frac{1}{2}}dx$
= $ \frac{x^{\frac{3}{2}}}{\frac{3}{2}} + \frac{1}{2}(\frac{x^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}) + C $
= $ \frac{2x^{\frac{3}{2}}}{3} + \frac{1}{2}(\frac{x^{\frac{1}{2}}}{\frac{1}{2}}) + C $
= $ \frac{2x^{\frac{3}{2}}}{3} + \frac{1}{2}(\frac{2x^{\frac{1}{2}}}{1}) + C $
= $ \frac{2}{3}x^{\frac{3}{2}} + \frac{2}{2}x^{\frac{1}{2}} + C $
= $ \frac{2}{3}x^{\frac{3}{2}} + x^{\frac{1}{2}} + C $
The result can be checked by differentiating, and it works.
