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.
Answer
\( \frac{2}{3} x^{3/2} + C \)
Answer for screen readers
The final answer is ( \frac{2}{3} x^{3/2} + C )
Steps to Solve
- Rewrite the function in exponent form
Rewrite $\sqrt{x}$ as $x^{1/2}$ because it's easier to work with exponents when integrating.
$$\int \sqrt{x} , dx = \int x^{1/2} , dx$$
- 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 $n = \frac{1}{2}$:
$$\int x^{1/2} , dx = \frac{1}{1/2 + 1}x^{1/2 + 1} + C$$
- 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 )
More Information
The antiderivative gives a family of functions that differ by a constant, all of which have the same derivative. This constant is represented by C.
Tips
A common mistake is to forget to add the constant of integration (C).