What is the antiderivative of 3 x^2?
Understand the Problem
The question is asking for the antiderivative of the function 3x^2, which involves finding a function whose derivative is 3x^2.
Answer
The antiderivative of $3x^2$ is $F(x) = x^3 + C$.
Answer for screen readers
The antiderivative of $3x^2$ is $F(x) = x^3 + C$.
Steps to Solve
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Identify the function The function we need to find the antiderivative for is $3x^2$.
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Apply the power rule for integration To find the antiderivative of $x^n$, we use the power rule: $$ \int x^n , dx = \frac{x^{n+1}}{n+1} + C $$ In our case, the constant $3$ can be factored out.
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Integrate the function We can rewrite the integral: $$ \int 3x^2 , dx = 3 \int x^2 , dx $$ Now apply the power rule: $$ = 3 \cdot \frac{x^{2+1}}{2+1} + C $$ $$ = 3 \cdot \frac{x^3}{3} + C $$
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Simplify the result Now simplify the expression: $$ = x^3 + C $$
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State the final antiderivative So, the antiderivative of $3x^2$ is: $$ F(x) = x^3 + C $$
The antiderivative of $3x^2$ is $F(x) = x^3 + C$.
More Information
When finding antiderivatives, it's essential to remember the constant of integration $C$. This constant represents any horizontal shift in the family of functions because the derivative of a constant is zero. As a result, multiple functions can have the same derivative if they differ by a constant.
Tips
- Forgetting to include the constant of integration $C$ after finding the antiderivative.
- Confusing the rules for integration and differentiation, leading to errors in the final expression.
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