The Indefinite Integral and Basic Integration Techniques
The big idea: integration is differentiation run backwards. Given a function's rate of change, the indefinite integral hands you back the whole family of functions that could have produced it — one antiderivative for every possible starting value.
- Read and unpack integral notation like a native speaker of calculus.
- Use linearity and the basic integral table fluently, term by term.
- Rewrite integrands algebraically so a table formula applies.
- Run substitution as a disciplined, repeatable ritual — not a guess.
- Recognize the patterns that collapse to logarithms, arctangent and arcsine.
- Verify any answer instantly, by differentiating it back.
A function $F$ is an antiderivative of $f$ on an interval if $F'(x) = f(x)$ for every $x$ in that interval. If $F$ is one antiderivative, then so is $F(x)+C$ for any constant $C$, because the derivative of a constant is zero. It's a theorem (not a coincidence) that on a connected interval these are the only antiderivatives — two functions with the same derivative differ by a constant. The indefinite integral is defined to be this entire family:
$$\int f(x)\,dx = F(x) + C, \qquad \text{where } F'(x) = f(x)$$
Every piece of this notation carries meaning: