antiderivative of x^(2/3)
Understand the Problem
The question is asking for the antiderivative of the function x^(2/3). To find the antiderivative, we will apply the power rule of integration, which states that the integral of x^n is (x^(n+1))/(n+1) plus a constant of integration.
Answer
The antiderivative of $x^{\frac{2}{3}}$ is $ \frac{3}{5} x^{\frac{5}{3}} + C $.
Answer for screen readers
The antiderivative of the function $x^{\frac{2}{3}}$ is
$$ \frac{3}{5} x^{\frac{5}{3}} + C $$
Steps to Solve
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Identify the function and the power
We are working with the function $x^{\frac{2}{3}}$.
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Apply the power rule of integration
According to the power rule, the antiderivative is calculated as follows:
$$ \int x^n , dx = \frac{x^{n+1}}{n+1} + C $$
In our case, $n = \frac{2}{3}$. Therefore, we will add 1 to the exponent:
$$ n + 1 = \frac{2}{3} + 1 = \frac{2}{3} + \frac{3}{3} = \frac{5}{3} $$
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Calculate the antiderivative
Now we will insert the new exponent into the power rule formula:
$$ \int x^{\frac{2}{3}} , dx = \frac{x^{\frac{5}{3}}}{\frac{5}{3}} + C $$
Dividing by a fraction is the same as multiplying by its reciprocal:
$$ \frac{x^{\frac{5}{3}}}{\frac{5}{3}} = x^{\frac{5}{3}} \cdot \frac{3}{5} = \frac{3}{5} x^{\frac{5}{3}} $$
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Add the constant of integration
Finally, we will include the constant of integration $C$:
$$ \int x^{\frac{2}{3}} , dx = \frac{3}{5} x^{\frac{5}{3}} + C $$
The antiderivative of the function $x^{\frac{2}{3}}$ is
$$ \frac{3}{5} x^{\frac{5}{3}} + C $$
More Information
This result shows the application of the power rule of integration. The constant $C$ represents an infinite number of antiderivatives that differ by a constant value. Understanding this concept is crucial in calculus, especially when dealing with indefinite integrals.
Tips
- Forgetting to add the constant of integration $C$ after finding the antiderivative.
- Miscalculating the exponent when applying the power rule, especially when dealing with fractions.
