How to solve this indefinite integral

Question

I'm trying to understand how to solve this indefinite integral using the power rule.

$$ \begin{eqnarray} \int \frac{x}{\sqrt{x}} dx &=& \int \frac{x^1}{x^\frac{1}{2}} dx \end{eqnarray} $$

Breaking this down

$$ \int x dx = \frac{x^2}{2} + C $$

$$ \begin{eqnarray} \int x^\frac{1}{2} dx &=& \frac{x^\frac{3}{2}}{\frac{3}{2}} + C\\ &=& \frac{2}{3} x^\frac{3}{2} + C \end{eqnarray} $$

Wolfram Alpha gives the integral of the original denominator ($\int\sqrt{x} dx$) as the solution. I do not understand what happened to the numerator?

How do I bring the two parts together?

Answer 1

$$\frac{x}{\sqrt{x}} = \frac{\sqrt{x}\sqrt{x}}{\sqrt{x}} = \sqrt{x}$$

$$\int \sqrt{x}\ dx = \frac{2}{3}x^{3/2} + c$$

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Note that $\sqrt{x} = x^{1/2}$ hence when you integrate it, just apply the integration rule for $x^a$.

$$\int x^a\ dx = \frac{x^{a+1}}{a+1} + c ~~~~~~~~~~~ \text{for} ~~~ a\neq -1$$

Answer 2

Hint: You can simplify $x\over \sqrt x$ into just $\sqrt x$. Then integrate:

$$\int \sqrt x \, dx$$

By the reverse power rule.