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{{Short description|Relationship between derivatives and integrals}}
{{Calculus}}

The '''fundamental theorem of calculus''' is a [[Theorem|theorem]] that links the concept of [[Derivative|differentiating]] a [[Function (mathematics)|function]] (calculating its [[Slope|slope]]s, or rate of change at each time) with the concept of [[Integral|integrating]] a function (calculating the area under its graph, or the cumulative effect of small contributions). The two operations are inverses of each other apart from a constant value which depends on where one starts to compute area.

The first part of the theorem, the '''first fundamental theorem of calculus''', states that for a function {{mvar|f}} , an [[Antiderivative|antiderivative]] or indefinite integral {{mvar|F}} may be obtained as the integral of {{mvar|f}} over an interval with a variable upper bound. This implies the existence of antiderivatives for [[Continuous function|continuous function]]s.<ref name=Spivak>{{Citation |last=Spivak|first=Michael|year=1980|title=Calculus|edition=2nd|publication-place=Houston, Texas| publisher=Publish or Perish Inc.}}</ref>

Conversely, the second part of the theorem, the '''second fundamental theorem of calculus''', states that the integral of a function {{mvar|f}} over a fixed [[Interval (mathematics)|interval]] is equal to the change of any antiderivative {{mvar|F}} between the ends of the interval. This greatly simplifies the calculation of a definite integral provided an antiderivative can be found by [[Symbolic integration|symbolic integration]], thus avoiding [[Numerical integration|numerical integration]].


==History==

The fundamental theorem of calculus relates differentiation and integration, showing that these two operations are essentially inverses of one another. Before the discovery of this theorem, it was not recognized that these two operations were related. Ancient Greek mathematicians knew how to compute area via infinitesimals, an operation that we would now call integration. The origins of differentiation likewise predate the fundamental theorem of calculus by hundreds of years; for example, in the fourteenth century the notions of ''[[Continuous function|continuity]]'' of functions and ''[[Physics:Motion|motion]]'' were studied by the [[Physics:Oxford Calculators|Oxford Calculators]] and other scholars. The historical relevance of the fundamental theorem of calculus is not the ability to calculate these operations, but the realization that the two seemingly distinct operations (calculation of geometric areas, and calculation of gradients) are actually closely related.

From the conjecture and the proof of the fundamental theorem of calculus, [[Calculus|calculus]] as a unified theory of integration and differentiation is started. The first published statement and proof of a rudimentary form of the fundamental theorem, strongly geometric in character,<ref name=GregoryGeometry>{{cite journal| last=Malet|first=Antoni|title=James Gregorie on tangents and the "Taylor" rule for series expansions| publisher=[[Company:Springer Science+Business Media|Springer-Verlag]]|year=1993|journal=Archive for History of Exact Sciences|volume=46|issue=2|pages=97–137|doi=10.1007/BF00375656|s2cid=120101519|quote=Gregorie's thought, on the other hand, belongs to a conceptual framework strongly geometrical in character. (page 137)}}</ref> was by [[Biography:James Gregory (mathematician)|James Gregory]] (1638–1675).<ref>
See, e.g., Marlow Anderson, Victor J. Katz, Robin J. Wilson, ''Sherlock Holmes in Babylon and Other Tales of Mathematical History'', Mathematical Association of America, 2004, [https://books.google.com/books?id=BKRE5AjRM3AC&pg=PA114 p.&nbsp;114].
</ref><ref name=geometriae>{{cite book| last=Gregory | first=James | title=Geometriae Pars Universalis | url=https://archive.org/details/gregory_universalis | publisher= Patavii: typis heredum Pauli Frambotti | year=1668 | location=Museo Galileo }}</ref> Isaac Barrow (1630–1677) proved a more generalized version of the theorem,<ref name=BarrowGeometricLectures>{{cite book| last1=Child | first1= James Mark | last2=Barrow | first2=Isaac| title= The Geometrical Lectures of Isaac Barrow | year=1916 | url=https://archive.org/details/geometricallectu00barruoft| publisher= Chicago: [[Company:Open Court Publishing Company|Open Court Publishing Company]]}}</ref> while his student [[Biography:Isaac Newton|Isaac Newton]] (1642–1727) completed the development of the surrounding mathematical theory. Gottfried Leibniz (1646–1716) systematized the knowledge into a calculus for infinitesimal quantities and introduced [[Leibniz's notation|the notation]] used today.

==Geometric meaning==
[[File:FTC_geometric.svg|500px|thumb|right|The area shaded in red stripes is close to {{mvar|h}} times {{math|''f''(''x'')}}. Alternatively, if the function {{math|''A''(''x'')}} were known, this area would be exactly {{math|''A''(''x'' + ''h'') − ''A''(''x'')}}. These two values are approximately equal, particularly for small {{mvar|h}}.]]
The first fundamental theorem may be interpreted as follows. Given a [[Continuous function|continuous function]] {{math|1=''y'' = ''f''(''x'')}} whose graph is plotted as a curve, one defines a corresponding "area function" <math>x\mapsto A(x)</math> such that {{math|''A''(''x'')}} is the area beneath the curve between {{math|0}} and {{mvar|x}}. The area {{math|''A''(''x'')}} may not be easily computable, but it is assumed to be well defined.

The area under the curve between {{mvar|x}} and {{math|''x'' + ''h''}} could be computed by finding the area between {{math|0}} and {{math|''x'' + ''h''}}, then subtracting the area between {{math|0}} and {{mvar|x}}. In other words, the area of this "strip" would be {{math|''A''(''x'' + ''h'') − ''A''(''x'')}}.

There is another way to ''estimate'' the area of this same strip. As shown in the accompanying figure, {{mvar|h}} is multiplied by {{math|''f''(''x'')}} to find the area of a rectangle that is approximately the same size as this strip. So:
<math display="block">A(x+h)-A(x) \approx f(x) \cdot h</math>

In fact, this estimate becomes a perfect equality if we add the red "Excess" area in the diagram. So:
<math display="block">A(x+h)-A(x)=f(x)\cdot h+\text{ (Excess)}</math>

Rearranging terms:
<math display="block">f(x) = \frac{A(x+h)-A(x)}{h} - \frac{\text{Excess}}{h}.</math>

As {{mvar|h}} approaches {{math|0}} in the [[Limit of a function|limit]], the last fraction must go to zero.<ref>[[Biography:Lipman Bers|Bers, Lipman]]. ''Calculus'', pp.&nbsp;180–181 (Holt, Rinehart and Winston (1976).</ref> To see this, note that the excess region is inside the tiny black-bordered rectangle, giving an upper bound for the excess area:<blockquote><math>|\text{Excess}| \le  h\,(f(x{+}h_1) - f(x{+}h_2)),</math></blockquote>where <math>x+h_1</math> and <math>x + h_2</math> are points where {{mvar|f}}  reaches its maximum and its minimum, respectively, in the interval {{closed-closed|''x'', ''x'' + ''h''}}.

Thus:
<math display="block">\left|f(x) - \frac{A(x+h) - A(x)}{h}\right| 
= \frac{|\text{Excess}|}{h} \le \frac{h(f(x+h_1) - f(x+h_2))}{h} 
= f(x{+}h_1) - f(x{+}h_2),</math>By the continuity of {{mvar|f}}, the right-hand expression tends to zero as {{mvar|h}} does. Therefore, the left-hand side also tends to zero, and:

<math display="block">f(x) = \lim_{h\to 0}\frac{A(x+h)-A(x)}{h}
\ \stackrel{\text{def}}{=}\ A'(x).</math>

That is, the derivative of the area function {{math|''A''(''x'')}} exists and is equal to the original function {{math|''f''(''x'')}}, so the area function is an [[Antiderivative|antiderivative]] of the original function.

Thus, the derivative of the integral of a function (the area) is the original function, so that derivative and integral are [[Inverse function|inverse operations]] which reverse each other. This is the essence of the Fundamental Theorem.

==Physical intuition==
Intuitively, the fundamental theorem states that ''integration'' and ''differentiation'' are essentially inverse operations which reverse each other.

The second fundamental theorem says that the sum of [[Infinitesimal|infinitesimal]] changes in a quantity over time (the integral of the derivative of the quantity) adds up to the net change in the quantity. To visualize this, imagine traveling in a car and wanting to know the distance traveled (the net change in position along the highway). You can see the velocity on the speedometer but cannot look out to see your location. Each second, you can find how far the car has traveled using {{math|1=distance = speed × time}}, multiplying the current speed (in kilometers or miles per hour) by the time interval (1 second = <math>\tfrac{1}{3600}</math> hour). Summing up all these small steps, you can calculate the ''total'' distance traveled, without ever looking outside the car:<math display="block">\text{distance traveled} 
= \sum \left( \begin{array}{c}
\text{velocity at}\\ \text{each time}\end{array}\right) \times 
\left( \begin{array}{c}
\text{time}\\ \text{interval}\end{array}\right)
= \sum v(t)\times \Delta t.</math>As <math>\Delta t</math> becomes [[Infinitesimal|infinitesimal]]ly small, the summing up corresponds to [[Integral|integration]]. Thus, the integral of the velocity function (the derivative of position) computes how far the car has traveled (the net change in position).

The first fundamental theorem says that any quantity is the rate of change (the derivative) of the integral of the quantity from a fixed time up to a variable time. Continuing the above example, if you imagine a velocity function, you can integrate it from the starting time up to any given time to obtain a distance function whose derivative is the given velocity. (To obtain the highway-marker position, you need to add your starting position to this integral.)

==Formal statements==
There are two parts to the theorem. The first part deals with the derivative of an [[Antiderivative|antiderivative]], while the second part deals with the relationship between antiderivatives and definite integrals.

===First part===
This part is sometimes referred to as the ''first fundamental theorem of calculus''.<ref>{{harvnb|Apostol|1967|loc=§5.1}}</ref>

Let {{mvar|f}} be a continuous [[Real-valued function|real-valued function]] defined on a closed interval {{closed-closed|''a'', ''b''}}. Let {{mvar|F}} be the function defined, for all {{mvar|x}} in {{closed-closed|''a'', ''b''}}, by
<math display="block">F(x) = \int_a^x f(t)\, dt.</math>

Then {{mvar|F}} is uniformly continuous on {{math|[''a'', ''b'']}} and differentiable on the open interval {{open-open|''a'', ''b''}}, and
<math display="block">F'(x) = f(x)</math>
for all {{mvar|x}} in {{open-open|''a'', ''b''}} so {{mvar|F}} is an antiderivative of {{mvar|f}}.

===Corollary===
[[File:Fundamental_theorem_of_calculus_(animation_).gif|thumb|Fundamental theorem of calculus (animation)]]

The fundamental theorem is often employed to compute the definite integral of a function <math>f</math> for which an antiderivative <math>F</math> is known. Specifically, if <math>f</math> is a real-valued continuous function on <math>[a,b]</math> and <math>F</math> is an antiderivative of <math>f</math> in <math>[a,b]</math>, then
<math display="block">\int_a^b f(t)\, dt = F(b)-F(a).</math>

The corollary assumes [[Continuous function|continuity]] on the whole interval. This result is strengthened slightly in the following part of the theorem.

===Second part===
This part is sometimes referred to as the ''second fundamental theorem of calculus''<ref>{{harvnb|Apostol|1967|loc=§5.3}}</ref> or the '''Newton–Leibniz theorem'''.

Let <math>f</math> be a real-valued function on a closed interval <math>[a,b]</math> and <math>F</math> a continuous function on <math>[a,b]</math> which is an antiderivative of <math>f</math> in <math>(a,b)</math>:
<math display="block">F'(x) = f(x).</math>

If <math>f</math> is Riemann integrable on <math>[a,b]</math> then
<math display="block">\int_a^b f(x)\,dx = F(b) - F(a).</math>

The second part is somewhat stronger than the corollary because it does not assume that <math>f</math> is continuous.

When an antiderivative <math>F</math> of <math>f</math> exists, then there are infinitely many antiderivatives for <math>f</math>, obtained by adding an arbitrary constant to <math>F</math>. Also, by the first part of the theorem, antiderivatives of <math>f</math> always exist when <math>f</math> is continuous.

==Proof of the first part==
For a given function {{math|''f''}}, define the function {{math|''F''(''x'')}} as
<math display="block">F(x) = \int_a^x f(t) \,dt.</math>

For any two numbers {{math|''x''<sub>1</sub>}} and {{math|''x''<sub>1</sub> + Δ''x''}} in {{closed-closed|''a'', ''b''}}, we have

<math display=block>\begin{align}F(x_1 + \Delta x) - F(x_1) &= \int_a^{x_1 + \Delta x} f(t) \,dt - \int_a^{x_1} f(t) \,dt
\\& =\int_{x_1}^{x_1 + \Delta x} f(t) \,dt, \end{align}</math>
the latter equality resulting from the basic properties of integrals and the additivity of areas.

According to the [[Mean value theorem#First mean value theorem for definite integrals|mean value theorem for integration]], there exists a real number <math>c \in [x_1, x_1 + \Delta x]</math> such that
<math display="block">\int_{x_1}^{x_1 + \Delta x} f(t) \,dt = f(c)\cdot \Delta x.</math>

It follows that
<math display="block">F(x_1 + \Delta x) - F(x_1) = f(c)\cdot \Delta x,</math>
and thus that
<math display="block">\frac{F(x_1 + \Delta x) - F(x_1)}{\Delta x} = f(c).</math>

Taking the limit as <math>\Delta x \to 0,</math> and keeping in mind that <math>c \in [x_1, x_1 + \Delta x],</math> one gets 
<math display="block">\lim_{\Delta x \to 0} \frac{F(x_1 + \Delta x) - F(x_1)}{\Delta x} = \lim_{\Delta x \to 0} f(c), </math>
that is, 
<math display="block">F'(x_1) = f(x_1),</math>
according to the definition of the derivative, the continuity of {{mvar|f}}, and the [[Squeeze theorem|squeeze theorem]].<ref>{{citation|last=Leithold|first=L.|year=1996|title=The calculus of a single variable|edition=6th| publication-place=New York|publisher=HarperCollins College Publishers|page=380}}.</ref>

==Proof of the corollary==
Suppose {{mvar|F}} is an antiderivative of {{mvar|f}}, with {{mvar|f}} continuous on {{closed-closed|''a'', ''b''}}. Let
<math display="block">G(x) = \int_a^x f(t)\, dt.</math>

By the ''first part'' of the theorem, we know {{mvar|G}} is also an antiderivative of {{mvar|f}}. Since {{math|1=''F''′ − ''G''′ = 0}} the [[Mean value theorem|mean value theorem]] implies that {{math|''F'' − ''G''}} is a [[Constant function|constant function]], that is, there is a number {{mvar|c}} such that {{math|1=''G''(''x'') = ''F''(''x'')&thinsp;+&thinsp;''c''}} for all {{mvar|x}} in {{closed-closed|''a'', ''b''}}. Letting {{math|1=''x'' = ''a''}}, we have
<math display="block">F(a) + c = G(a) = \int_a^a f(t)\, dt = 0,</math>
which means {{math|1=''c'' = −''F''(''a'')}}. In other words, {{math|1=''G''(''x'') = ''F''(''x'') − ''F''(''a'')}}, and so
<math display="block">\int_a^b f(x)\, dx = G(b) = F(b) - F(a).</math>

==Proof of the second part==
This is a limit proof by [[Riemann integral|Riemann sums]].

To begin, we recall the [[Mean value theorem|mean value theorem]]. Stated briefly, if {{mvar|F}} is continuous on the closed interval {{closed-closed|''a'', ''b''}} and differentiable on the open interval {{open-open|''a'', ''b''}}, then there exists some {{mvar|c}} in {{open-open|''a'', ''b''}} such that
<math display="block">F'(c)(b - a) = F(b) - F(a). </math>

Let {{mvar|f}} be (Riemann) integrable on the interval {{closed-closed|''a'', ''b''}}, and let {{mvar|f}} admit an antiderivative {{mvar|F}} on {{open-open|''a'', ''b''}} such that {{mvar|F}} is continuous on {{closed-closed|''a'', ''b''}}. Begin with the quantity {{math|''F''(''b'') − ''F''(''a'')}}. Let there be numbers {{math|''x''<sub>0</sub>, ..., ''x''<sub>''n''</sub>}} such that
<math display="block">a = x_0 < x_1 < x_2 < \cdots < x_{n-1} < x_n = b. </math>

It follows that
<math display="block">F(b) - F(a) = F(x_n) - F(x_0). </math>

Now, we add each {{math|''F''(''x''<sub>''i''</sub>)}} along with its additive inverse, so that the resulting quantity is equal:
<math display="block">\begin{align}
F(b) - F(a)
&= F(x_n) + [-F(x_{n-1}) + F(x_{n-1})] + \cdots + [-F(x_1) + F(x_1)] - F(x_0) \\
&= [F(x_n) - F(x_{n-1})] + [F(x_{n-1}) - F(x_{n-2})] + \cdots + [F(x_2) - F(x_1)] + [F(x_1) - F(x_0)].
\end{align}</math>

The above quantity can be written as the following sum:
{{NumBlk||<math display="block">F(b) - F(a) = \sum_{i=1}^n [F(x_i) - F(x_{i-1})].</math>|{{EquationRef|1'}}}}

The function {{mvar|F}} is differentiable on the interval {{open-open|''a'', ''b''}} and continuous on the closed interval {{closed-closed|''a'', ''b''}}; therefore, it is also differentiable on each interval {{open-open|''x''<sub>''i''−1</sub>, ''x''<sub>''i''</sub>}} and continuous on each interval {{closed-closed|''x''<sub>''i''−1</sub>, ''x''<sub>''i''</sub>}}. According to the mean value theorem (above), for each {{mvar|i}} there exists a <math>c_i</math> in {{open-open|''x''<sub>''i''−1</sub>, ''x''<sub>''i''</sub>}} such that
<math display="block">F(x_i) - F(x_{i-1}) = F'(c_i)(x_i - x_{i-1}). </math>

Substituting the above into ({{EquationNote|1'}}), we get
<math display="block">F(b) - F(a) = \sum_{i=1}^n [F'(c_i)(x_i - x_{i-1})].</math>

The assumption implies <math>F'(c_i) = f(c_i).</math> Also, <math>x_i - x_{i-1}</math> can be expressed as <math>\Delta x</math> of partition <math>i</math>.
{{NumBlk||<math display="block">F(b) - F(a) = \sum_{i=1}^n [f(c_i)(\Delta x_i)]. </math>|{{EquationRef|2'}}}}

[[File:Riemann integral irregular.gif|frame|right|A converging sequence of Riemann sums. The number in the upper left is the total area of the blue rectangles. They converge to the definite integral of the function.]]

We are describing the area of a rectangle, with the width times the height, and we are adding the areas together. Each rectangle, by virtue of the [[Mean value theorem|mean value theorem]], describes an approximation of the curve section it is drawn over. Also <math>\Delta x_i</math> need not be the same for all values of {{mvar|i}}, or in other words that the width of the rectangles can differ. What we have to do is approximate the curve with {{mvar|n}} rectangles. Now, as the size of the partitions get smaller and {{mvar|n}} increases, resulting in more partitions to cover the space, we get closer and closer to the actual area of the curve.

By taking the limit of the expression as the norm of the partitions approaches zero, we arrive at the [[Riemann integral]]. We know that this limit exists because {{mvar|f}} was assumed to be integrable. That is, we take the limit as the largest of the partitions approaches zero in size, so that all other partitions are smaller and the number of partitions approaches infinity.

So, we take the limit on both sides of ({{EquationNote|2'}}). This gives us
<math display="block">\lim_{\| \Delta x_i \| \to 0} F(b) - F(a) = \lim_{\| \Delta x_i \| \to 0} \sum_{i=1}^n [f(c_i)(\Delta x_i)].</math>

Neither {{math|''F''(''b'')}} nor {{math|''F''(''a'')}} is dependent on <math>\|\Delta x_i\|</math>, so the limit on the left side remains {{math|''F''(''b'') − ''F''(''a'')}}.
<math display="block">F(b) - F(a) = \lim_{\| \Delta x_i \| \to 0} \sum_{i=1}^n [f(c_i)(\Delta x_i)].</math>

The expression on the right side of the equation defines the integral over {{mvar|f}} from {{mvar|a}} to {{mvar|b}}. Therefore, we obtain
<math display="block">F(b) - F(a) = \int_a^b f(x)\,dx,</math>
which completes the proof.

==Relationship between the parts==

As discussed above, a slightly weaker version of the second part follows from the first part.

Similarly, it almost looks like the first part of the theorem follows directly from the second. That is, suppose {{mvar|G}} is an antiderivative of {{mvar|f}}. Then by the second theorem, <math display="inline">G(x) - G(a) = \int_a^x f(t) \, dt</math>. Now, suppose <math display="inline">F(x) = \int_a^x f(t)\, dt = G(x) - G(a)</math>. Then {{mvar|F}} has the same derivative as {{mvar|G}}, and therefore {{math|1=''F''′ = ''f''}}. This argument only works, however, if we already know that {{mvar|f}} has an antiderivative, and the only way we know that all continuous functions have antiderivatives is by the first part of the Fundamental Theorem.<ref name=Spivak/>
For example, if {{math|1=''f''(''x'') = ''e''<sup>−''x''<sup>2</sup></sup>}}, then {{mvar|f}} has an antiderivative, namely
<math display="block">G(x) = \int_0^x f(t) \, dt</math>
and there is no simpler expression for this function. It is therefore important not to interpret the second part of the theorem as the definition of the integral. Indeed, there are many [[Nonelementary integral|functions that are integrable but lack elementary antiderivatives]], and discontinuous functions can be integrable but lack any antiderivatives at all. Conversely, many functions that have antiderivatives are not Riemann integrable (see [[Volterra's function]]).

==Examples==
===Computing a particular integral===
Suppose the following is to be calculated:
<math display="block">\int_2^5 x^2\, dx. </math>

Here, <math>f(x) = x^2 </math> and we can use <math display="inline">F(x) = \frac{1}{3}x^3 </math> as the antiderivative. Therefore:
<math display="block">\int_2^5 x^2\, dx = F(5) - F(2) = \frac{5^3}{3} - \frac{2^3}{3} = \frac{125}{3} - \frac{8}{3} = \frac{117}{3} = 39.</math>

===Using the first part===
Suppose
<math display="block">
\frac{d}{dx} \int_0^x t^3\, dt
</math>
is to be calculated. Using the first part of the theorem with <math>f(t) = t^3 </math> gives
<math display="block">
\frac{d}{dx} \int_0^x t^3\, dt = f(x)= x^3.
</math>

This can also be checked using the second part of the theorem. Specifically, <math display="inline">F(t) = \frac{1}{4}t^4 </math> is an antiderivative of <math>f(t)</math>, so
<math display="block">
\frac{d}{dx} \int_0^x t^3\, dt = \frac{d}{dx} F(x) - \frac{d}{dx} F(0) = \frac{d}{dx} \frac{x^4}{4} = x^3.
</math>

===An integral where the corollary is insufficient===
Suppose
<math display="block">
f(x)=\begin{cases} \sin\left(\frac1x\right)-\frac1x\cos\left(\frac1x\right) & x\ne0\\
0 & x=0\\
\end{cases}
</math>
Then <math>f(x)</math> is not continuous at zero. Moreover, this is not just a matter of how <math>f</math> is defined at zero, since the limit as <math>x\to0</math> of <math>f(x)</math> does not exist. Therefore, the corollary cannot be used to compute
<math display="block">
\int_0^1 f(x)\, dx.
</math>
But consider the function
<math display="block">
F(x)=\begin{cases} x\sin\left(\frac1x\right) & x\ne0\\
0 & x=0.\\
\end{cases}
</math>
Notice that <math>F(x)</math> is continuous on <math>[0,1]</math> (including at zero by the [[Squeeze theorem|squeeze theorem]]), and <math>F(x)</math> is differentiable on <math>(0,1)</math> with <math>F'(x)=f(x).</math> Therefore, part two of the theorem applies, and
<math display="block">
\int_0^1 f(x)\, dx=F(1)-F(0)=\sin(1).
</math>

===Theoretical example===
The theorem can be used to prove that
<math display="block">\int_a^b f(x) dx = \int_a^c f(x) dx+\int_c^b f(x) dx.</math>

Since, 
<math display="block"> \begin{align}
\int_a^b f(x) dx &= F(b)-F(a), \\
\int_a^c f(x) dx &= F(c)-F(a), \text{ and } \\
\int_c^b f(x) dx &= F(b)-F(c), 
\end{align}</math>
the result follows from, 
<math display="block">F(b)-F(a) = F(c)-F(a)+F(b)-F(c).</math>

==Generalizations==
The function {{mvar|f}} does not have to be continuous over the whole interval. Part I of the theorem then says: if {{mvar|f}} is any [[Lebesgue integration|Lebesgue integrable]] function on {{closed-closed|''a'', ''b''}} and {{math|''x''<sub>0</sub>}} is a number in {{closed-closed|''a'', ''b''}} such that {{mvar|f}} is continuous at {{math|''x''<sub>0</sub>}}, then
<math display="block">F(x) = \int_a^x f(t)\, dt</math>

is differentiable for {{math|1=''x'' = ''x''<sub>0</sub>}} with {{math|1=''F''′(''x''<sub>0</sub>) = ''f''(''x''<sub>0</sub>)}}. We can relax the conditions on {{mvar|f}} still further and suppose that it is merely locally integrable. In that case, we can conclude that the function {{mvar|F}} is differentiable [[Almost everywhere|almost everywhere]] and {{math|1=''F''′(''x'') = ''f''(''x'')}} almost everywhere. On the [[Real line|real line]] this statement is equivalent to [[Lebesgue differentiation theorem|Lebesgue's differentiation theorem]]. These results remain true for the [[Henstock–Kurzweil integral]], which allows a larger class of integrable functions.{{sfnp|Bartle|2001|loc=Thm. 4.11}}

In higher dimensions Lebesgue's differentiation theorem generalizes the Fundamental theorem of calculus by stating that for almost every {{mvar|x}}, the average value of a function {{mvar|f}} over a ball of radius {{mvar|r}} centered at {{mvar|x}} tends to {{math|''f''(''x'')}} as {{mvar|r}} tends to 0.

Part II of the theorem is true for any Lebesgue integrable function {{mvar|f}}, which has an antiderivative {{mvar|F}} (not all integrable functions do, though). In other words, if a real function {{mvar|F}} on {{closed-closed|''a'', ''b''}} admits a derivative {{math|''f''(''x'')}} at ''every'' point {{mvar|x}} of {{closed-closed|''a'', ''b''}} and if this derivative {{mvar|f}} is Lebesgue integrable on {{closed-closed|''a'', ''b''}}, then<ref>{{harvnb|Rudin|1987|loc=th. 7.21}}</ref>
<math display="block">F(b) - F(a) = \int_a^b f(t) \, dt.</math>

This result may fail for continuous functions {{mvar|F}} that admit a derivative {{math|''f''(''x'')}} at almost every point {{mvar|x}}, as the example of the [[Cantor function]] shows. However, if {{mvar|F}} is [[Absolute continuity|absolutely continuous]], it admits a derivative {{math|''F′''(''x'')}} at almost every point {{mvar|x}}, and moreover {{mvar|F′}} is integrable, with {{math|''F''(''b'') − ''F''(''a'')}} equal to the integral of {{mvar|F′}} on {{closed-closed|''a'', ''b''}}. Conversely, if {{mvar|f}} is any integrable function, then {{mvar|F}} as given in the first formula will be absolutely continuous with {{math|1=''F′'' = ''f''}} almost everywhere.

The conditions of this theorem may again be relaxed by considering the integrals involved as [[Henstock–Kurzweil integral]]s. Specifically, if a continuous function {{math|''F''(''x'')}} admits a derivative {{math|''f''(''x'')}} at all but countably many points, then {{math|''f''(''x'')}} is Henstock–Kurzweil integrable and {{math|''F''(''b'') − ''F''(''a'')}} is equal to the integral of {{mvar|f}} on {{closed-closed|''a'', ''b''}}. The difference here is that the integrability of {{mvar|f}} does not need to be assumed.{{sfnp|Bartle|2001|loc=Thm. 4.7}}

The version of [[Taylor's theorem]], which expresses the error term as an integral, can be seen as a generalization of the fundamental theorem.

There is a version of the theorem for [[Complex number|complex]] functions: suppose {{mvar|U}} is an [[Open set|open set]] in {{math|'''C'''}} and {{math|''f'' : ''U'' → '''C'''}} is a function that has a [[Holomorphic function|holomorphic]] antiderivative {{mvar|F}} on {{mvar|U}}. Then for every curve {{math|γ : [''a'', ''b''] → ''U''}}, the curve integral can be computed as
<math display="block">\int_\gamma f(z) \,dz = F(\gamma(b)) - F(\gamma(a)).</math>

The fundamental theorem can be generalized to curve and surface integrals in higher dimensions and on [[Manifold|manifold]]s. One such generalization offered by the [[Calculus of moving surfaces|calculus of moving surfaces]] is the [[Time evolution of integrals|time evolution of integrals]]. The most familiar extensions of the fundamental theorem of calculus in higher dimensions are the [[Divergence theorem|divergence theorem]] and the [[Gradient theorem|gradient theorem]].

One of the most powerful generalizations in this direction is [[Generalized Stokes theorem|Stokes' theorem]] (sometimes known as the fundamental theorem of multivariable calculus):<ref>{{Cite book|title-link=Calculus on Manifolds (book)|title=Calculus on Manifolds| last=Spivak|first=M.|publisher=W. A. Benjamin|year=1965|isbn=978-0-8053-9021-6|location=New York|pages=124–125}}</ref> Let {{mvar|M}} be an oriented [[Piecewise|piecewise]] smooth [[Manifold|manifold]] of [[Physics:Dimension|dimension]] {{mvar|n}} and let <math>\omega</math> be a smooth compactly supported [[Differential form|{{math|(''n''&thinsp;−&thinsp;1)}}-form]] on {{mvar|M}}. If {{math|∂''M''}} denotes the [[Manifold|boundary]] of {{mvar|M}} given its induced orientation, then
<math display="block">\int_M d\omega = \int_{\partial M} \omega.</math>

Here {{math|''d''}} is the [[Exterior derivative|exterior derivative]], which is defined using the manifold structure only.

The theorem is often used in situations where {{mvar|M}} is an embedded oriented submanifold of some bigger manifold (e.g. {{math|'''R'''<sup>''k''</sup>}}) on which the form <math>\omega</math> is defined.

The fundamental theorem of calculus allows us to pose a definite integral as a first-order ordinary differential equation.
<math display="block">\int_a^b f(x)\, dx</math>
can be posed as
<math display="block">\frac{dy}{dx}=f(x),\;\; y(a)=0 </math>
with <math>y(b)</math> as the value of the integral.

==See also==
* Differentiation under the integral sign
* [[Telescoping series]]
* [[Gradient theorem|Fundamental theorem of calculus for line integrals]]
* [[Notation for differentiation]]

==Notes==
{{reflist|group=Note}}

==References==
{{Reflist|30em}}

===Bibliography===
{{refbegin}}
* {{Citation | last1=Apostol | first1=Tom M. |  title=Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra | publisher={{wipe|John Wiley & Sons}} | location=New York | edition=2nd | isbn=978-0-471-00005-1 | year=1967 | url-access=registration | url=https://archive.org/details/calculus01apos }}.
* {{Citation | last1=Bartle | first1=Robert | title=A Modern Theory of Integration | publisher=AMS | isbn=0-8218-0845-1 | year=2001}}.
* {{citation|last=Leithold|first=L.|year=1996|title=The calculus of a single variable|edition=6th|publication-place=New York|publisher=HarperCollins College Publishers}}.
* {{citation|last=Rudin|first=Walter|year=1987|title=Real and Complex Analysis|edition=third| publication-place=New York|publisher=McGraw-Hill Book Co.|isbn=0-07-054234-1 }}
{{refend}}

==Further reading==
{{refbegin}}
* {{citation|last1=Courant|first1=Richard|last2=John|first2=Fritz|title=Introduction to Calculus and Analysis|publisher=Springer|year=1965}}.
* {{citation|last1=Larson|first1=Ron|first2=Bruce H.|last2=Edwards|first3=David E.|last3=Heyd|title=Calculus of a single variable|edition=7th| isbn=978-0-618-14916-2 | publication-place=Boston|publisher=Houghton Mifflin Company|year=2002}}.
* [[Biography:Antoni Malet|Malet, A.]], ''Studies on James Gregorie (1638-1675)'' (PhD Thesis, Princeton, 1989).
* Hernandez Rodriguez, O. A.; Lopez Fernandez, J. M. . "[http://www.maa.org/publications/periodicals/convergence/teaching-the-fundamental-theorem-of-calculus-a-historical-reflection-introduction Teaching the Fundamental Theorem of Calculus: A Historical Reflection]", ''Loci: Convergence'' (MAA), January 2012.
* {{citation|last=Stewart|first=J.|year=2003|contribution=Fundamental Theorem of Calculus|title=Calculus: early transcendentals| publication-place=Belmont, California|publisher=Thomson/Brooks/Cole}}.
* {{citation|editor=Turnbull, H. W.|title=The James Gregory Tercentenary Memorial Volume|publication-place=London|year=1939}}.
{{refend}}

==External links==
* {{springer|title=Fundamental theorem of calculus|id=p/n066550}}
* [https://web.archive.org/web/20070715022739/http://mathdl.maa.org/convergence/1/?pa=content&sa=viewDocument&nodeId=388&bodyId=343 James Gregory's Euclidean Proof of the Fundamental Theorem of Calculus] at Convergence
* [http://school.maths.uwa.edu.au/~schultz/L18Barrow.html Isaac Barrow's proof of the Fundamental Theorem of Calculus]
* [https://imomath.com/index.cgi?page=calculusDefiniteIntegralsFundamentalTheoremOfCalculus Fundamental Theorem of Calculus at imomath.com]
* [https://web.archive.org/web/20160613164124/http://www.proofs.wiki/Fundamental_theorem_of_calculus Alternative proof of the fundamental theorem of calculus]
* [https://ocw.mit.edu/courses/18-01sc-single-variable-calculus-fall-2010/pages/unit-3-the-definite-integral-and-its-applications/part-a-definition-of-the-definite-integral-and-first-fundamental-theorem/session-47-introduction-of-the-fundamental-theorem-of-calculus/ Fundamental Theorem of Calculus] MIT.
* [http://mathworld.wolfram.com/FundamentalTheoremsofCalculus.html Fundamental Theorem of Calculus] Mathworld.

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