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Integral see math notation, function, antiderivative, calculus notation notation in conventional math notation with `dd F x -- dd x = f`, \(F\ x\ \bigr\|_{a}^{b} \dots = F\ b \cdot F\ a\) definition a definite integral has its two endpoints present definition an indefinite integral has its two endpoints missing definition the integrand is the function being integrated. it would be `f x` in the integral `$ f x \| dd x` Integration antiderivative › antidifferentiation see antiderivative › antidifferentiation Improper Integration type I an integral with at least one endpoint being infinite theorem `$ f x \| dd x {@@ . a} == $ f x \| dd x {t . a} {t -> @@}` theorem `$ f x \| dd x {b . .@@} == $ f x \| dd x {b . t} {t -> .@@}` a Type I improper integral is said to: converge if the limit exists diverge if the limit does not exist type II an integral whose integrand has a function discontinuity on the integration interval theorem if `f x {x -> b "from the left"} = {@@ \/ .@@}` #xxx improved expression evaluation, then `$ f x \| dd x {b . a} == $ f x \| dd x {t . a} {t -> b "from the left"}` theorem if `f x {x -> a "from the right"} = {@@ \/ .@@}` #xxx improved expression evaluation, `$ f x \| dd x {b . a} == $ f x \| dd x {b . t} {t -> a "from the right"}` comparison test theorem Comparison Test let `0 -\| g x -\| f x` on an interval *`a {-\|/\+} {-\|/\+} b`, where `a` and `b` are not necessarily finite. then, if `$ f x \| dd x {b . a}` converges, so does `$ g x \| dd x {b . a}`, but not conversely if `$ g x \| dd x {b . a}` diverges, so does `$ f x \| dd x {b . a}`, but not conversely p-test --- https://youtu.be/TKIdC847K3k theorem** P-Test the integral `$ --[x]p \| dd x {@@ . 1}`: converges if `p {\|-/\+} 1` diverges if `p -\| 1` Numerical Integration the definite integral of a function can be estimated by the monte carlo method by generating a finite number of random points and calculating the ratio between the number of points under the function and the total number of points --- https://youtu.be/rUxP7TM8-wo?t=2925. the accuracy of such an algorithm can be measured through inferential statistics
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Integral

see math notation, function, antiderivative, calculus notation

notation in conventional math notation

with dd F x -- dd x = f,

$F\ x\ \bigr|_{a}^{b} \dots = F\ b \cdot F\ a$

definition a definite integral has its two endpoints present

definition an indefinite integral has its two endpoints missing

definition the integrand is the function being integrated. it would be f x in the integral $ f x | dd x

Integration

antiderivative › antidifferentiation

see antiderivative › antidifferentiation

Improper Integration

type I

an integral with at least one endpoint being infinite

theorem $ f x | dd x {@@ . a} == $ f x | dd x {t . a} {t -> @@}

theorem $ f x | dd x {b . .@@} == $ f x | dd x {b . t} {t -> .@@}

a Type I improper integral is said to:

converge if the limit exists
diverge if the limit does not exist

type II

an integral whose integrand has a function discontinuity on the integration interval

theorem if f x {x -> b "from the left"} = {@@ \/ .@@} #xxx improved expression evaluation, then $ f x | dd x {b . a} == $ f x | dd x {t . a} {t -> b "from the left"}

theorem if f x {x -> a "from the right"} = {@@ \/ .@@} #xxx improved expression evaluation, $ f x | dd x {b . a} == $ f x | dd x {b . t} {t -> a "from the right"}

comparison test

theorem Comparison Test

let 0 -| g x -| f x on an interval a {-|/\+} * {-|/\+} b, where a and b are not necessarily finite. then,

if $ f x | dd x {b . a} converges, so does $ g x | dd x {b . a}, but not conversely
if $ g x | dd x {b . a} diverges, so does $ f x | dd x {b . a}, but not conversely

p-test

--- https://youtu.be/TKIdC847K3k

theorem P-Test

the integral $ --[x]p | dd x {@@ . 1}:

converges if p {|-/\+} 1
diverges if p -| 1

Numerical Integration

the definite integral of a function can be estimated by the monte carlo method by generating a finite number of random points and calculating the ratio between the number of points under the function and the total number of points --- https://youtu.be/rUxP7TM8-wo?t=2925. the accuracy of such an algorithm can be measured through inferential statistics