Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation AB & BC

$F$ is an antiderivative of $f$ when $F' = f$, and there is never just one: any two differ by a constant, and every constant occurs. The indefinite integral $\int f(x)\,dx = F(x) + C$ names that whole family, which makes it a family of functions rather than a number, and separates it from the definite integral, where any constant appears in both substitutions and cancels. The reverse power rule raises the exponent and divides by the new one, $\int x^{n}dx = \frac{x^{n+1}}{n+1} + C$, and it excludes $n = -1$ because that would divide by zero; the antiderivative of $\frac{1}{x}$ is $\ln|x| + C$, with bars because $\frac{1}{x}$ is defined for negative $x$.

The standard list is the derivative rules read backwards, with the sign landing on $\int \sin x\,dx = -\cos x + C$. When the inside is linear a reciprocal coefficient comes out front, so $\int e^{2x}dx = \frac{1}{2}e^{2x} + C$, because differentiating would otherwise leave a stray factor of $2$; a non-linear inside gets no such treatment. An initial condition turns the family into one function: antidifferentiate keeping $+C$, substitute the point, solve, and write the value in. Every answer in this topic can be checked in seconds by differentiating it back.