# [[Antiderivatives]] (Mathematics > Calculus) ## Definition An antiderivative of a function is a function whose derivative is the original function. It is the reverse process of differentiation. For any function $f(x)$, an antiderivative $F(x)$ satisfies: $ F'(x) = f(x) $ For example, the antiderivative of $2x$ is $x^2$ because the derivative of $x^2$ is $2x$. ## Key Concepts - **Indefinite Integral**: The family of all antiderivatives of a function, represented with an arbitrary constant $C$. - **Power Rule for Antiderivatives**: If $f(x) = x^n$, then an antiderivative is $F(x) = \frac{x^{n+1}}{n+1}$ (for $n \neq -1$). - **Arbitrary Constant**: Since differentiation of constants yields zero, any antiderivative has a constant of integration, $C$. ## Important Properties 1. The antiderivative is not unique; it includes a constant of integration, $C$. 2. Antiderivatives are the inverse operation of derivatives. 3. Every continuous function has an infinite number of antiderivatives. ## Essential Formulas - General form of an antiderivative: $ \int f(x) \, dx = F(x) + C $ - Power rule: $ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad \text{for } n \neq -1 $ ## Core Examples 1. The antiderivative of $2x$ is: $ \int 2x \, dx = x^2 + C $ 2. The antiderivative of $3x^2$ is: $ \int 3x^2 \, dx = x^3 + C $ ## Related Theorems/Rules - **Fundamental Theorem of Calculus**: Links antiderivatives with definite integrals. - **Linearity of Integration**: $\int [a f(x) + b g(x)] \, dx = a \int f(x) \, dx + b \int g(x) \, dx$ ## Common Pitfalls - Forgetting to include the constant of integration $C$. - Confusing the antiderivative with the definite integral, which involves limits of integration. ## Related Topics - [[Derivative]] - [[Power Rule of Limits]] - [[Integral]] - [[Fundamental Theorem of Calculus]] - [[Definite Integral]] - [[Fundamental Theorem of Calculus]] ## Quick Review Questions 1. What is the antiderivative of $5x^4$? 2. How does the antiderivative differ from a definite integral? *** ![[Antiderivatives_visualization]]