Antiderivatives: Rules, Definite Integrals, and Existence

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What is an antiderivative?

A differentiable function $F$ is an antiderivative of $f$ if $F'(x)=f(x)$ on the interval being considered.

What does an indefinite integral represent?

It represents the family of antiderivatives of a function, conventionally written with the constant of integration $C$.

Why is the constant of integration included?

Because differentiating any constant gives zero, all functions that differ by a constant have the same derivative.

State the power rule for antiderivatives.

For $n\ne -1$, $\int x^n\,dx=\frac{x^{n+1}}{n+1}+C$.

What is the antiderivative of $1/x$?

$\ln|x|+C$. The absolute value allows for intervals where $x$ is positive or negative, while excluding zero.

Quiz questions

1. What is the difference between an indefinite integral and a definite integral?Answer: An indefinite integral represents a family of functions; a definite integral is a number representing signed accumulation.
2. In one-dimensional motion, what information determines the constant when acceleration is integrated to find velocity?Answer: The specified initial velocity.
3. What does continuity guarantee for a function defined on an interval?Answer: The function has an antiderivative on that interval.
4. If a function has domain consisting of two disconnected intervals, how may its antiderivatives differ between the intervals?Answer: They may have separate constants of integration on the two intervals.
5. Which statement about the antiderivative of $e^{-x^2}$ is correct?Answer: It exists but cannot be expressed using elementary functions.