If differentiation gives you the rate of change, an antiderivative takes you back to the original function. For example, if you know the velocity of a car, the antiderivative gives you its position. The set of all antiderivatives of a function is called the indefinite integral.
🎯 In this section you will learn
- The definition of an antiderivative
- Indefinite integral notation $\int f(x) dx = F(x) + C$
- Why we always add $+C$ (the constant of integration)
- Basic antiderivative formulas
📌 Definition of an Antiderivative
A function $F$ is an antiderivative of $f$ on an interval if $F'(x) = f(x)$ for all $x$ in that interval.
$$ F'(x) = f(x) \quad \Longleftrightarrow \quad \int f(x) \, dx = F(x) + C $$
The indefinite integral represents the family of all antiderivatives
💡 The Constant $C$
If $F$ is an antiderivative of $f$, then $F(x) + C$ is also an antiderivative for any constant $C$, because the derivative of a constant is $0$. This is called the constant of integration.
Find an antiderivative of $f(x) = 3x^2$.
①
Think backward
What function has derivative $3x^2$? The derivative of $x^3$ is $3x^2$.
Find all antiderivatives of $f(x) = \cos x$.
- Derivative of $\sin x$ is $\cos x$
- Derivative of $\sin x + 5$ is also $\cos x$
- All antiderivatives: $\sin x + C$
Find $\int e^x \, dx$.
- Derivative of $e^x$ is $e^x$, so $\int e^x dx = e^x + C$
Find $\int \frac{1}{x} \, dx$ for $x > 0$.
- Derivative of $\ln x$ is $\frac{1}{x}$, so $\int \frac{1}{x} dx = \ln x + C$ (for $x > 0$)
⚠️ Important Note
Antiderivatives are not unique. If $F$ is one antiderivative, then $F + C$ works for any constant $C$. That's why indefinite integrals always include $+C$.
🔍 Key Takeaways
- An antiderivative of $f$ is a function $F$ such that $F'(x) = f(x)$.
- The indefinite integral $\int f(x) dx = F(x) + C$ represents the entire family of antiderivatives.
- Always include the constant of integration $+C$ for indefinite integrals.
- Basic formulas: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$, $\int e^x dx = e^x + C$, $\int \frac{1}{x} dx = \ln|x| + C$.
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