# Integrate square root of x.

#### Understand the Problem

The question is asking for the integral of the square root of x with respect to x. To solve this, we will apply the power rule for integration.

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

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

#### 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:

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

1. 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:

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

1. 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:

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

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

When integrating, always remember to include the constant of integration, $C$.
A common mistake is to forget the constant of integration, $C$. Additionally, make sure to correctly simplify the exponents and fractions.