Give me integral anti-differentiation problems.
Understand the Problem
The question is asking for problems related to integral anti-differentiation, which involve finding the indefinite integrals of various functions.
Answer
The indefinite integral of \( f(x) = x^2 \) is $$ \int x^2 \, dx = \frac{x^3}{3} + C $$
Answer for screen readers
The indefinite integral of ( f(x) = x^2 ) is:
$$ \int x^2 , dx = \frac{x^3}{3} + C $$
Steps to Solve
- Identify the function to integrate
First, determine the function that you need to integrate. For example, let's consider the function ( f(x) = x^2 ).
- Apply the power rule for integration
Using the power rule of integration, you can integrate the function. The power rule states that $$ \int x^n , dx = \frac{x^{n+1}}{n+1} + C $$ where ( n \neq -1 ) and ( C ) is the constant of integration.
For our example function:
$$ \int x^2 , dx = \frac{x^{2+1}}{2+1} + C = \frac{x^3}{3} + C $$
- Add the constant of integration
Don't forget to add the constant of integration ( C ) at the end, as it represents any constant value that would differentiate to zero.
Thus, the complete indefinite integral for our example is:
$$ \int x^2 , dx = \frac{x^3}{3} + C $$
- Check your work
Verify your answer by differentiating your result. If you differentiate ( \frac{x^3}{3} + C ), you should return to the original function:
$$ \frac{d}{dx} \left( \frac{x^3}{3} + C \right) = x^2 $$
This confirms the integration is done correctly.
The indefinite integral of ( f(x) = x^2 ) is:
$$ \int x^2 , dx = \frac{x^3}{3} + C $$
More Information
Indefinite integrals are important in calculus for finding functions based on their rates of change. Integrating a function gives us not just one answer, but a family of answers represented by the constant ( C ).
Tips
- Forgetting to include the constant of integration ( C ) which is critical in indefinite integrals.
- Misapplying the power rule, especially with negative powers.
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