Integral
An integral is the inverse of a derivative; an antiderivative.
Description
A definite integral represents the area under a curve from a to b. It is notated as:
Fundamental to solving such integration problems is understanding that integrals are antiderivatives. Reference derivative rules and apply them in inverse. Concretely, letting F be the antiderivative of f, and assuming f is integrable on [a,b], the above can be solved as:
An improper integral sets an infinite bound, like:
Indefinite Integrals
An indefinite integral is the generalization of integration, notated as:
Because the derivative of a constant is 0, there are infinitely many antiderivatives for any given function. To represent the generalized set of antiderivatives, add a constant C.
Letting F be the antiderivative of f, an indefinite integral is solved as:
Vector-Valued Functions
The indefinite integral of a vector-valued function is given by:
Letting F, G, and H be the antiderivatives of f, g, and h respectively; and letting C be a vector of Ci, Cj, and Ck, this integral can be expressed as:
