Tag Info

Concerns all aspects of integration, including the integral definition and computational methods. For questions solely about the properties of integrals, use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that typically describe(s) the types of the integrals being considered. This tag often goes along with the (calculus) tag.

Integration is a major part of , related to differentiation by the Fundamental Theorem of Calculus, Part Two:

If $$f$$ is the derivative of $$F$$, then $$\int^b_a f(x)\ dx = F(b)-F(a).$$

Note: to be strict, the equality in the formula above is not always achieved, but requires $$f$$ to have certain properties. For example, $$f$$ is absolutely continuous and $$f$$ is the derivative of $$F$$ almost everywhere. However, for most functions encountered in real-world problems, the equality in the above formula is true.

There are two main kinds of integrals: definite integrals (e.g. proper and improper integrals), which often have numerical values, and indefinite integrals, which group families of functions with the same derivative.

Integration is often harder than differentiation. Several techniques have been developed, including integration by parts, substitution, trigonometric substitution, and partial fractions.

Integration can be used to find the area under a graph and find the average of the function. Also, it can be used to compute the volume of certain solids and to find the displacement of a particle.

There are many ways of formally defining an integral, not all of which are equivalent. The differences exist mostly to deal with differing special cases which may not be integrable under other definitions, but also occasionally for pedagogical reasons. The most commonly used definitions of the integral are Riemann integrals and Lebesgue integrals.