Question: The anti derivative of \begin{align} \left(\sqrt x + \frac {1}{\sqrt x}\right)\end{align} equals to \begin{align} (A) \frac{1}{3}.x^\frac{1}{3} + 2.x^\frac{1}{2} +C \;\;\;\; (B) \frac{2}{3}.x^\frac{2}{3} + \frac{1}{2}.x^2 +C \end{align}
\begin{align} (C) \frac{2}{3}.x^\frac{3}{2} +2 x^\frac{1}{2} +C \;\;\;\;(D) \frac{3}{2}.x^\frac{3}{2} +\frac{1}{2}. x^\frac{1}{2} +C \end{align}

Answer:

\begin{align} \int\left(\sqrt x + \frac {1}{\sqrt x}\right).dx \end{align}

\begin{align}= \int x^\frac{1}{2}.dx + \int x^\frac{-1}{2}.dx \end{align}

\begin{align}= \left(\frac {x^\frac{3}{2}}{\frac{3}{2}}\right) + \left(\frac {x^\frac{1}{2}}{\frac{1}{2}}\right) + C\end{align}

\begin{align}= \frac {2}{3} . x^{\frac{3}{2}}+ 2x^\frac{1}{2} + C\end{align}

Hence, the Correct Answer is C.

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