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==Properties==

===Linearity===
The collection of Riemann-integrable functions on a closed interval {{math|[''a'', ''b'']}} forms a [[vector space]] under the operations of [[pointwise addition]] and multiplication by a scalar, and the operation of integration

: <math> f \mapsto \int_a^b f(x) \; dx</math>

is a [[linear functional]] on this vector space. Thus, the collection of integrable functions is closed under taking [[linear combination]]s, and the integral of a linear combination is the linear combination of the integrals:<ref name=":0">{{Harvnb|Apostol|1967|p=80}}.</ref>

: <math> \int_a^b (\alpha f + \beta g)(x) \, dx = \alpha \int_a^b f(x) \,dx + \beta \int_a^b g(x) \, dx. \,</math>

Similarly, the set of [[Real number|real]]-valued Lebesgue-integrable functions on a given [[Measure (mathematics)|measure space]] {{mvar|E}} with measure {{mvar|μ}} is closed under taking linear combinations and hence form a vector space, and the Lebesgue integral

: <math> f\mapsto \int_E f \, d\mu </math>

is a linear functional on this vector space, so that:<ref name=":3" />

: <math> \int_E (\alpha f + \beta g) \, d\mu = \alpha \int_E f \, d\mu + \beta \int_E g \, d\mu. </math>

More generally, consider the vector space of all [[measurable function]]s on a measure space {{math|(''E'',''μ'')}}, taking values in a [[Locally compact space|locally compact]] [[Complete metric space|complete]] [[topological vector space]] {{mvar|V}} over a locally compact [[Topological ring|topological field]] {{math|''K'', ''f'' : ''E'' → ''V''}}. Then one may define an abstract integration map assigning to each function {{mvar|f}} an element of {{mvar|V}} or the symbol {{math|''∞''}},

: <math> f\mapsto\int_E f \,d\mu, \,</math>

that is compatible with linear combinations.<ref>{{Harvnb|Rudin|1987|p=54}}.</ref> In this situation, the linearity holds for the subspace of functions whose integral is an element of {{mvar|V}} (i.e. "finite"). The most important special cases arise when {{mvar|K}} is {{math|'''R'''}}, {{math|'''C'''}}, or a finite extension of the field {{math|'''Q'''''p''}} of [[p-adic number]]s, and {{mvar|V}} is a finite-dimensional vector space over {{mvar|K}}, and when {{math|''K'' {{=}} '''C'''}} and {{mvar|V}} is a complex [[Hilbert space]].

Linearity, together with some natural continuity properties and normalization for a certain class of "simple" functions, may be used to give an alternative definition of the integral. This is the approach of [[Daniell integral|Daniell]] for the case of real-valued functions on a set {{mvar|X}}, generalized by [[Nicolas Bourbaki]] to functions with values in a locally compact topological vector space. See {{Harvnb|Hildebrandt|1953}} for an axiomatic characterization of the integral.

=== Inequalities ===
A number of general inequalities hold for Riemann-integrable [[Function (mathematics)|functions]] defined on a [[Closed set|closed]] and [[Bounded set|bounded]] [[Interval (mathematics)|interval]] {{closed-closed|''a'', ''b''}} and can be generalized to other notions of integral (Lebesgue and Daniell).

* ''Upper and lower bounds.'' An integrable function {{mvar|f}} on {{closed-closed|''a'', ''b''}}, is necessarily [[Bounded function|bounded]] on that interval. Thus there are [[real number]]s {{mvar|m}} and {{mvar|M}} so that {{math|''m'' ≤ ''f'' (''x'') ≤ ''M''}} for all {{mvar|x}} in {{closed-closed|''a'', ''b''}}. Since the lower and upper sums of {{mvar|f}} over {{closed-closed|''a'', ''b''}} are therefore bounded by, respectively, {{math|''m''(''b'' − ''a'')}} and {{math|''M''(''b'' − ''a'')}}, it follows that <math display="block"> m(b - a) \leq \int_a^b f(x) \, dx \leq M(b - a). </math>
* ''Inequalities between functions.''<ref>{{Harvnb|Apostol|1967|p=81}}.</ref> If {{math|''f''(''x'') ≤ ''g''(''x'')}} for each {{mvar|x}} in {{closed-closed|''a'', ''b''}} then each of the upper and lower sums of {{mvar|f}} is bounded above by the upper and lower sums, respectively, of {{mvar|g}}. Thus <math display="block"> \int_a^b f(x) \, dx \leq \int_a^b g(x) \, dx. </math> This is a generalization of the above inequalities, as {{math|''M''(''b'' − ''a'')}} is the integral of the constant function with value {{mvar|M}} over {{closed-closed|''a'', ''b''}}. In addition, if the inequality between functions is strict, then the inequality between integrals is also strict. That is, if {{math|''f''(''x'') < ''g''(''x'')}} for each {{mvar|x}} in {{closed-closed|''a'', ''b''}}, then <math display="block"> \int_a^b f(x) \, dx < \int_a^b g(x) \, dx. </math>
* ''Subintervals.'' If {{closed-closed|''c'', ''d''}} is a subinterval of {{closed-closed|''a'', ''b''}} and {{math|''f''&hairsp;(''x'')}} is non-negative for all {{mvar|x}}, then <math display="block"> \int_c^d f(x) \, dx \leq \int_a^b f(x) \, dx. </math>
* ''Products and absolute values of functions.'' If {{mvar|f}} and {{mvar|g}} are two functions, then we may consider their [[pointwise product]]s and powers, and [[absolute value]]s: <math display="block">
(fg)(x)= f(x) g(x), \; f^2 (x) = (f(x))^2, \; |f| (x) = |f(x)|.</math> If {{mvar|f}} is Riemann-integrable on {{closed-closed|''a'', ''b''}} then the same is true for {{math|{{abs|''f''}}}}, and <math display="block">\left| \int_a^b f(x) \, dx \right| \leq \int_a^b | f(x) | \, dx. </math> Moreover, if {{mvar|f}} and {{mvar|g}} are both Riemann-integrable then {{math|''fg''}} is also Riemann-integrable, and <math display="block">\left( \int_a^b (fg)(x) \, dx \right)^2 \leq \left( \int_a^b f(x)^2 \, dx \right) \left( \int_a^b g(x)^2 \, dx \right). </math> This inequality, known as the [[Cauchy–Schwarz inequality]], plays a prominent role in [[Hilbert space]] theory, where the left hand side is interpreted as the [[Inner product space|inner product]] of two [[Square-integrable function|square-integrable]] functions {{mvar|f}} and {{mvar|g}} on the interval {{closed-closed|''a'', ''b''}}.
* ''Hölder's inequality''.<ref name=":4">{{Harvnb|Rudin|1987|p=63}}.</ref> Suppose that {{mvar|p}} and {{mvar|q}} are two real numbers, {{math|1 ≤ ''p'', ''q'' ≤ ∞}} with {{math|1={{sfrac|1|''p''}} + {{sfrac|1|''q''}} = 1}}, and {{mvar|f}} and {{mvar|g}} are two Riemann-integrable functions. Then the functions {{math|{{abs|''f''}}''p''}} and {{math|{{abs|''g''}}''q''}} are also integrable and the following [[Hölder's inequality]] holds: <math display="block">\left|\int f(x)g(x)\,dx\right| \leq
\left(\int \left|f(x)\right|^p\,dx \right)^{1/p} \left(\int\left|g(x)\right|^q\,dx\right)^{1/q}.</math> For {{math|1=''p'' = ''q'' = 2}}, Hölder's inequality becomes the Cauchy–Schwarz inequality.
* ''Minkowski inequality''.<ref name=":4" /> Suppose that {{math|''p'' ≥ 1}} is a real number and {{mvar|f}} and {{mvar|g}} are Riemann-integrable functions. Then {{math|{{abs| ''f'' }}''p'', {{abs| ''g'' }}''p''}} and {{math|{{abs| ''f'' + ''g'' }}''p''}} are also Riemann-integrable and the following [[Minkowski inequality]] holds: <math display="block">\left(\int \left|f(x)+g(x)\right|^p\,dx \right)^{1/p} \leq
\left(\int \left|f(x)\right|^p\,dx \right)^{1/p} +
\left(\int \left|g(x)\right|^p\,dx \right)^{1/p}.</math> An analogue of this inequality for Lebesgue integral is used in construction of [[Lp space|Lp spaces]].

=== Conventions ===
In this section, {{mvar|f}} is a [[Real number|real-valued]] Riemann-integrable [[Function (mathematics)|function]]. The integral

: <math> \int_a^b f(x) \, dx </math>

over an interval {{math|[''a'', ''b'']}} is defined if {{math|''a'' < ''b''}}. This means that the upper and lower sums of the function {{mvar|f}} are evaluated on a partition {{math|''a'' {{=}} ''x''0 ≤ ''x''1 ≤ . . . ≤ ''x''''n'' {{=}} ''b''}} whose values {{math|''x''''i''}} are increasing. Geometrically, this signifies that integration takes place "left to right", evaluating {{mvar|f}} within intervals {{math|[''x'' ''i'' , ''x'' ''i'' +1]}} where an interval with a higher index lies to the right of one with a lower index. The values {{mvar|a}} and {{mvar|b}}, the end-points of the [[Interval (mathematics)|interval]], are called the [[limits of integration]] of {{mvar|f}}. Integrals can also be defined if {{math|''a'' > ''b''}}:''<ref name=":1" />''

:<math>\int_a^b f(x) \, dx = - \int_b^a f(x) \, dx. </math>

With {{math|''a'' {{=}} ''b''}}, this implies:

:<math>\int_a^a f(x) \, dx = 0. </math>

The first convention is necessary in consideration of taking integrals over subintervals of {{math|[''a'', ''b'']}}; the second says that an integral taken over a degenerate interval, or a [[Point (geometry)|point]], should be [[0 (number)|zero]]. One reason for the first convention is that the integrability of {{mvar|f}} on an interval {{math|[''a'', ''b'']}} implies that {{mvar|f}} is integrable on any subinterval {{math|[''c'', ''d'']}}, but in particular integrals have the property that if {{mvar|c}} is any [[Element (mathematics)|element]] of {{math|[''a'', ''b'']}}, then:''<ref name=":0" />''

:<math> \int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx.</math>

With the first convention, the resulting relation

: <math>\begin{align}
\int_a^c f(x) \, dx &{}= \int_a^b f(x) \, dx - \int_c^b f(x) \, dx \\
&{} = \int_a^b f(x) \, dx + \int_b^c f(x) \, dx
\end{align}</math>

is then well-defined for any cyclic permutation of {{mvar|a}}, {{mvar|b}}, and {{mvar|c}}.

=== Improper integrals ===
{{Main|Improper integral}}
[[File:Improper_integral.svg|right|thumb|The [[improper integral]]<math>\int_{0}^{\infty} \frac{dx}{(x+1)\sqrt{x}} = \pi</math>

has unbounded intervals for both domain and range.]]
A "proper" Riemann integral assumes the integrand is defined and finite on a closed and bounded interval, bracketed by the limits of integration. An improper integral occurs when one or more of these conditions is not satisfied. In some cases such integrals may be defined by considering the [[Limit (mathematics)|limit]] of a [[sequence]] of proper [[Riemann integral]]s on progressively larger intervals.

If the interval is unbounded, for instance at its upper end, then the improper integral is the limit as that endpoint goes to infinity:<ref>{{Harvnb|Apostol|1967|p=416}}.</ref>

: <math>\int_a^\infty f(x)\,dx = \lim_{b \to \infty} \int_a^b f(x)\,dx.</math>

If the integrand is only defined or finite on a half-open interval, for instance {{math|<nowiki>(</nowiki>''a'', ''b''<nowiki>]</nowiki>}}, then again a limit may provide a finite result:<ref>{{Harvnb|Apostol|1967|p=418}}.</ref>

: <math>\int_a^b f(x)\,dx = \lim_{\varepsilon \to 0} \int_{a+\epsilon}^{b} f(x)\,dx.</math>

That is, the improper integral is the [[Limit (mathematics)|limit]] of proper integrals as one endpoint of the interval of integration approaches either a specified [[real number]], or {{math|∞}}, or {{math|−∞}}. In more complicated cases, limits are required at both endpoints, or at interior points.

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 == Licensing ==
 Content obtained and/or adapted from:
-* [https://en.wikipedia.org/wiki/Integral Integral Wikipedia] under a CC BY-SA license
+* [https://en.wikipedia.org/wiki/Integral Integral, Wikipedia] under a CC BY-SA license

← Older revision		Revision as of 21:10, 26 October 2021
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	That is, the improper integral is the limit of proper integrals as one endpoint of the interval of integration approaches either a specified real number, or {{math\|∞}}, or {{math\|−∞}}. In more complicated cases, limits are required at both endpoints, or at interior points.		That is, the improper integral is the limit of proper integrals as one endpoint of the interval of integration approaches either a specified real number, or {{math\|∞}}, or {{math\|−∞}}. In more complicated cases, limits are required at both endpoints, or at interior points.
		+
		+	== Licensing ==
		+	Content obtained and/or adapted from:
		+	* [https://en.wikipedia.org/wiki/Integral Integral Wikipedia] under a CC BY-SA license

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 * ''Upper and lower bounds.'' An integrable function {{mvar|f}} on [''a'', ''b''], is necessarily bounded on that interval. Thus there are real numbers {{mvar|m}} and {{mvar|M}} so that {{math|''m'' ≤ ''f'' (''x'') ≤ ''M''}} for all {{mvar|x}} in [''a'', ''b'']. Since the lower and upper sums of {{mvar|f}} over [''a'', ''b''] are therefore bounded by, respectively, {{math|''m''(''b'' − ''a'')}} and {{math|''M''(''b'' − ''a'')}}, it follows that <math display="block"> m(b - a) \leq \int_a^b f(x) \, dx \leq M(b - a). </math>
 * ''Inequalities between functions.'' If {{math|''f''(''x'') ≤ ''g''(''x'')}} for each {{mvar|x}} in [''a'', ''b''] then each of the upper and lower sums of {{mvar|f}} is bounded above by the upper and lower sums, respectively, of {{mvar|g}}. Thus <math display="block"> \int_a^b f(x) \, dx \leq \int_a^b g(x) \, dx. </math> This is a generalization of the above inequalities, as {{math|''M''(''b'' − ''a'')}} is the integral of the constant function with value {{mvar|M}} over [''a'', ''b'']. In addition, if the inequality between functions is strict, then the inequality between integrals is also strict. That is, if {{math|''f''(''x'') < ''g''(''x'')}} for each {{mvar|x}} in [''a'', ''b''], then <math display="block"> \int_a^b f(x) \, dx < \int_a^b g(x) \, dx. </math>
-* ''Subintervals.'' If [''c'', ''d''] is a subinterval of [''a'', ''b''] and {{math|''f''&hairsp;(''x'')}} is non-negative for all {{mvar|x}}, then <math display="block"> \int_c^d f(x) \, dx \leq \int_a^b f(x) \, dx. </math>
+* ''Subintervals.'' If [''c'', ''d''] is a subinterval of [''a'', ''b''] and {{math|''f''(''x'')}} is non-negative for all {{mvar|x}}, then <math display="block"> \int_c^d f(x) \, dx \leq \int_a^b f(x) \, dx. </math>
 * ''Products and absolute values of functions.'' If {{mvar|f}} and {{mvar|g}} are two functions, then we may consider their pointwise products and powers, and absolute values: <math display="block">
   (fg)(x)= f(x) g(x), \; f^2 (x) = (f(x))^2, \; |f| (x) = |f(x)|.</math> If {{mvar|f}} is Riemann-integrable on [''a'', ''b''] then the same is true for |''f''|, and <math display="block">\left| \int_a^b f(x) \, dx \right| \leq \int_a^b | f(x) | \, dx. </math> Moreover, if {{mvar|f}} and {{mvar|g}} are both Riemann-integrable then {{math|''fg''}} is also Riemann-integrable, and <math display="block">\left( \int_a^b (fg)(x) \, dx \right)^2 \leq \left( \int_a^b f(x)^2 \, dx \right) \left( \int_a^b g(x)^2 \, dx \right). </math> This inequality, known as the Cauchy–Schwarz inequality, plays a prominent role in Hilbert space theory, where the left hand side is interpreted as the inner product of two square-integrable functions {{mvar|f}} and {{mvar|g}} on the interval [''a'', ''b''].