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.
Answer
$\frac{2}{3} x^{3/2} + C$
Answer for screen readers
The final answer is $$ \int \sqrt{x} , dx = \frac{2}{3} x^{3/2} + C $$
Steps to Solve
- Rewrite the square root as an exponent
The square root of $x$ can be written as $x^{1/2}$. So, we rewrite the integral:
$$ \int \sqrt{x} , dx = \int x^{1/2} , dx $$
- 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:
$$ \int x^{1/2} , dx = \frac{x^{1/2 + 1}}{1/2 + 1} + C $$
- 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 $$ \int \sqrt{x} , dx = \frac{2}{3} x^{3/2} + C $$
More Information
When integrating, always remember to include the constant of integration, $C$.
Tips
A common mistake is to forget the constant of integration, $C$. Additionally, make sure to correctly simplify the exponents and fractions.