Question

Write the primitive or anti-derivative of

\[f\left( x \right) = \sqrt{x} + \frac{1}{\sqrt{x}} .\]

 

Answer 1

` f (x) = \sqrtx + 1/ \sqrtx `.

integrating both sides

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