← Integration methods
1 Reading the notation
Every part of an integral sign carries information. Here is what each piece means.📖 Notation used in integration
9 termsCover the right column and check you can say what each one means.
- ∫
- The integral sign, an elongated S, for “sum”. Integration is fundamentally adding up infinitely many infinitesimal pieces.
- dx (at the end)
- Says which variable you are integrating with respect to, and marks the end of the integrand. Never optional.
- ∫ₐᵇ (limits)
- A definite integral, evaluated from a to b. The result is a NUMBER.
- ∫ with no limits
- An indefinite integral. The result is a FAMILY of functions, so it needs “+ c”.
- + c
- The constant of integration. Differentiating any constant gives zero, so infinitely many functions share the same derivative. A boundary condition (a known point) picks out which one you want.
- F(x) vs f(x)
- Capital F conventionally denotes the antiderivative of lower-case f. So F′(x) = f(x).
- [F(x)]ₐᵇ
- Notation for “evaluate F at b, then subtract F at a”. Equals F(b) − F(a).
- u (in substitution)
- A temporary variable standing for an inner function, chosen so the integral simplifies. du = u′(x) dx.
- |x| (absolute value)
- Needed in ∫(1/x)dx = ln|x| + c because logarithms are undefined for negative arguments, but 1/x is perfectly integrable there.