AcademyIntegration
Academy
Antiderivatives
Level 1 - Math I (Physics) topic page in Integration.
Antiderivatives and Indefinite Integrals
An antiderivative (also called a primitive function) of a function \(f(x)\) is a function \(F(x)\) such that \(F'(x) = f(x)\). The process of finding antiderivatives is called integration.
The Indefinite Integral
The indefinite integral notation uses the integral symbol \(\int\):
Indefinite Integral
\[\int f(x) \, dx = F(x) + C\]
where:
- \(\int\) is the integral symbol
- \(f(x)\) is the integrand (the function being integrated)
- \(dx\) indicates the variable of integration
- \(F(x)\) is an antiderivative of \(f(x)\)
- \(C\) is the constant of integration
Constant of Integration
When we differentiate \(F(x) + C\), the constant \(C\) disappears since the derivative of a constant is zero. Therefore, every antiderivative differs by a constant. This is why we always include \(+ C\).
All Antiderivatives
\[\text{If } F'(x) = f(x), \text{ then } \int f(x) \, dx = F(x) + C \text{ for any constant } C\]
Basic Power Rule for Integration
For \(n \neq -1\):
Power Rule
\[\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\]
Examples
- \(\int x^3 \, dx = \frac{x^4}{4} + C\)
- \(\int 5 \, dx = 5x + C\)
- \(\int \frac{1}{x} \, dx = \ln|x| + C\) (note the absolute value)
Verification
Always verify by differentiating your result:
Verification
\[\frac{d}{dx}\left(\frac{x^{n+1}}{n+1} + C\right) = x^n\]
The constant of integration ensures we capture all possible antiderivatives, not just one specific solution.