Linking differentiation and integration.
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Click on an arrow to get a description of the connection!
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Concept 
Content 
Differentiability 
How to quantify the rate of change of a function. 
Examples for
Calculating the
Riemann Integral 
Use the approximation by step functions to calculate integrals. 
Maps 
Maps are the mathematical formulation of a machine that gets inputs and generate outputs. On both sides, sets are needed for the domain and the codomain. 
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The next result now states that integrals can be determined by
inversion of differentiation.
Theorem 1 (Fundamental theorem of differentiation
and integration). Let \(I\) be an
interval and a continuous \(f:I\to\mathbb{R}\) be given. Let \(F:I\to\mathbb{R}\) be an antiderivative of
\(f\). Then for all \(a,b\in I\) holds \[\int_a^bf(x)\, dx=F(b)F(a).\] We write
\[\int_a^bf(x)\, dx=\left.
F(x)\right_{x=a}^{x=b}.\]
The above result gives rise to the following notation for an
antiderivative: \[\int f(x)\,
dx:=F(x).\] Based on our knowledge about differentiation, we now
collect some antiderivatives of important functions in the following
table.
Some antiderivatives


\(\displaystyle x^n,\quad
n\in\mathbb{N}\) 
\(\displaystyle \frac1{n+1}x^{n+1}\) 
\(\displaystyle x^{1},\quad x\neq0\) 
\(\displaystyle \log(x)\) 
\(\displaystyle x^{n},\quad x\neq0,n\in\mathbb{N},
n\neq1\) 
\(\displaystyle \frac1{1n}x^{1n}\) 
\(\displaystyle \exp(x)\) 
\(\displaystyle \exp(x)\) 
\(\displaystyle \sinh(x)\) 
\(\displaystyle \cosh(x)\) 
\(\displaystyle \cosh(x)\) 
\(\displaystyle \sinh(x)\) 
\(\displaystyle \frac1{\sqrt{1+x^2}}\) 
\(\displaystyle\operatorname{arsinh}(x)\) 
\(\displaystyle \frac1{\sqrt{x^21}},\quad
x>1\) 
\(\displaystyle\operatorname{arcosh}(x)\) 
\(\displaystyle \frac1{1x^2},\quad
x<1\) 
\(\displaystyle\operatorname{artanh}(x)\) 
\(\displaystyle \sin(x)\) 
\(\displaystyle\cos(x)\) 
\(\displaystyle \cos(x)\) 
\(\displaystyle\sin(x)\) 
\(\displaystyle
\frac1{\cos^2(x)}=1+\tan^2(x)\) 
\(\displaystyle \tan(x)\) 
\(\displaystyle
\frac1{\sqrt{1x^2}},\quadx<1\) 
\(\displaystyle \arcsin(x)\) 
\(\displaystyle
\frac1{\sqrt{1x^2}},\quadx<1\) 
\(\displaystyle \arccos(x)\) 
\(\displaystyle \frac1{1+x^2}\) 
\(\displaystyle \arctan(x)\) 
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