# [[Indefinite Integrals]] (Mathematics > Calculus) ## Definition An indefinite integral represents the family of antiderivatives of a function. It is expressed with an arbitrary constant $C$ because the derivative of a constant is zero. The indefinite integral of a function $f(x)$ is denoted as: $ \int f(x) \, dx = F(x) + C $ where $F(x)$ is the antiderivative of $f(x)$. ## Key Concepts - **Indefinite Integral**: Represents all antiderivatives of a function. - **Power Rule for Integration**: A method to integrate power functions. - **Constant of Integration**: The arbitrary constant $C$ added to the antiderivative. ## Important Properties 1. Every indefinite integral includes an arbitrary constant $C$. 2. The power rule does not apply when $n = -1$ (for integrals involving $\frac{1}{x}$). 3. Indefinite integrals represent families of functions. ## Essential Formulas - Power rule for indefinite integrals: $ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad \text{for} \, n \neq -1 $ - For $n = -1$, the integral is: $ \int \frac{1}{x} \, dx = \ln|x| + C $ ## Core Examples 1. The indefinite integral of $x^3$: $ \int x^3 \, dx = \frac{x^4}{4} + C $ 2. The indefinite integral of $\frac{1}{x}$: $ \int \frac{1}{x} \, dx = \ln|x| + C $ ## Related Theorems/Rules - **Linearity of Integration**: $ \int [a f(x) + b g(x)] \, dx = a \int f(x) \, dx + b \int g(x) \, dx $ ## Common Pitfalls - Forgetting the constant of integration $C$. - Misapplying the power rule when $n = -1$. ## Related Topics - [[Derivative]] - [[Indefinite Integrals]] - [[Power Rule of Limits]] - [[Logarithm]] - [[Natural Logarithm]] - [[Common Logarithm]] - [[Antiderivatives]] - [[Definite Integrals]] ## Quick Review Questions 1. What is the indefinite integral of $x^5$? 2. Why can't the power rule be used to find $\int \frac{1}{x} \, dx$? *** ![[Indefinite_Integrals_visualization]]