Integrate square root of x.

Steps to Solve

1. Rewrite the square root as an exponent

The square root of $x$ can be written as $x^{1/2}$. So, we rewrite the integral:

$$ \int \sqrt{x} , dx = \int x^{1/2} , dx $$

2. Apply the power rule for integration

The power rule for integration states that if you have an integral of the form $\int x^n , dx$, the result is $rac{x^{n+1}}{n+1} + C$, where $C$ is the constant of integration and $n \neq -1$. In our case, $n = \frac{1}{2}$.

Applying the power rule:

$$ \int x^{1/2} , dx = \frac{x^{1/2 + 1}}{1/2 + 1} + C $$

3. Simplify the expression

Simplify the exponent and the denominator:

$$ \frac{x^{3/2}}{3/2} + C = \frac{2}{3} x^{3/2} + C $$

The integral of $\sqrt{x}$ with respect to $x$ is:

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