Integrals
Understand the Problem
The question is asking for information related to integrals, which are fundamental concepts in calculus that involve finding the area under a curve or the accumulation of quantities.
Answer
The integral $\int x^2 \, dx$ is $\frac{x^3}{3} + C$.
Answer for screen readers
The solution to the integral $\int x^2 , dx$ is
$$ \frac{x^3}{3} + C $$
Steps to Solve
- Identify the Integral to Evaluate
Determine the function you need to integrate. Let's say we have $f(x) = x^2$. The integral to evaluate would be:
$$ \int f(x) , dx = \int x^2 , dx $$
- Apply the Power Rule for Integration
To solve the integral, use the power rule, which states that:
$$ \int x^n , dx = \frac{x^{n+1}}{n+1} + C $$
For our example, with $n=2$, this becomes:
$$ \int x^2 , dx = \frac{x^{2+1}}{2+1} + C = \frac{x^3}{3} + C $$
- Add the Constant of Integration
Remember to add the constant of integration $C$ to the result since integrals represent a family of functions. The final result in our example would be:
$$ \int x^2 , dx = \frac{x^3}{3} + C $$
The solution to the integral $\int x^2 , dx$ is
$$ \frac{x^3}{3} + C $$
More Information
This integral represents the area under the curve $f(x) = x^2$ from a specified range (if limits were provided) or the general antiderivative, which includes a constant of integration $C$ that accounts for all possible vertical shifts of the function.
Tips
- Forgetting to add the constant of integration $C$.
- Misapplying the power rule by not increasing the exponent or the denominator appropriately.
AI-generated content may contain errors. Please verify critical information
