# antiderivative of sqrt x

#### Understand the Problem

The question is asking for the antiderivative of the function sqrt(x), which is a mathematical operation that finds a function whose derivative is sqrt(x). The approach involves using integration rules.

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

The final answer is ( \frac{2}{3} x^{3/2} + C )

#### Steps to Solve

1. Rewrite the function in exponent form

Rewrite $\sqrt{x}$ as $x^{1/2}$ because it's easier to work with exponents when integrating.

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

1. Apply the power rule of integration

The power rule for integration states that $\int x^n , dx = \frac{1}{n+1}x^{n+1} + C$ where $n \neq -1$.

For $n = \frac{1}{2}$:

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

1. Simplify the expression

Simplify the exponent and the fraction:

$$\frac{1}{1/2 + 1} = \frac{1}{3/2} = \frac{2}{3}$$

Thus, $$x^{1/2 + 1} = x^{3/2}$$

So the integral becomes:

$$\int x^{1/2} , dx = \frac{2}{3} x^{3/2} + C$$

The final answer is ( \frac{2}{3} x^{3/2} + C )