OpenLLM-France/wikimedia huggingface dataset:
# MyOpenMath/Complex phasors
## Time average of the product of two signals

Here we multiply two signals with the same angular frequency but with different phases:
    $I(t)=I_{0}\cos(\omega t+\phi _{i})={\tfrac {1}{2}}I_{0}\exp {i(\omega t+\phi _{i})}+cc$
    $V(t)=V_{0}\cos(\omega t+\phi _{v})={\tfrac {1}{2}}V_{0}\exp {i(\omega t+\phi _{v})}+cc$,
where $cc$ denotes complex conjugate. For example,
$e^{i\theta }+cc=e^{i\theta }+e^{-i\theta }=2\cos \theta .$
Define $P=IV$, and make the algebra easier to follow by defining two phases:
    $\Phi _{I}=\omega t+\phi _{i},$ and, $\Phi _{V}=\omega t+\phi _{v}$.
Note that $PV$ is the product of two binomials, which yield four terms:
    $IV={\tfrac {1}{4}}I_{0}V_{0}\left(e^{i\Phi _{I}}+e^{-i\Phi _{I}}\right)\left(e^{i\Phi _{V}}+e^{-i\Phi _{V}}\right)$
When the two binomials are multiplied we obtain four terms. We group them according to whether they involve the sum or difference between the two phases, $\Phi _{I}$ and $\Phi _{V}$, because whether it is a sum or difference affects the time-dependence as follows:
    $\Phi _{I}+\Phi _{V}=2\omega t+\phi _{i}+\phi _{v}$
    $\Phi _{I}-\Phi _{V}=\phi _{i}-\phi _{v}$
These terms can be grouped into real and imaginary parts, expressed in terms of the sine and cosine functions:
    $IV={\tfrac {1}{4}}I_{0}V_{0}{\Bigl\underbrace {e^{i(\Phi _{I}-\Phi _{V})}+e^{i(\Phi _{V}-\Phi _{I})}} _{2\cos(\Phi _{I}-\Phi _{V})}{\Bigr ]}+{\Bigl(\underbrace {e^{i(\Phi _{I}+\Phi _{V})}+e^{-i(\Phi _{V}+\Phi _{V})}} _{2\cos(\Phi _{I}+\Phi _{V})}{\Bigr ]}$
With ac circuits it is customary to average over one period, $T$, defined by the expression $\omega T=2\pi$. Using the overbar notation to denote this time average, we have:
    ${\overline {IV}}={\tfrac {1}{2}}I_{0}V_{0}\cos(\phi _{i}-\phi _{v})=I_{\text{rms}}V_{\text{rms}}\cos(\phi _{i}-\phi _{v})$

## Footnotes

* ↑ We may also average over an integral number of periods. Or, with minimal error, we may simply average over any interval of time much greater than T.
# MyOpenMath/Solutions

Special:Prefixindex/MyOpenMath/Solutions/
# PlanetPhysics/Morita Uniqueness Theorem

The main result for Morita equivalent algebras is provided by the following proposition.
\begin{theorem}Morita theorem.
Let $A$ and $B$ be two arbitrary rings, and also let $F:A-mod\to B-mod$ be an additive, right exact functor. Then, there is a $(B,A)$-bimodule ${\mathcal {Q}}$, which is unique up to isomorphism, so that $F$ is isomorphic to the functor $G$ given by $A-mod\mapsto B-mod,$ $M\mapsto Q\bigotimes_{A}M.$ \end{theorem}
There are also two important and fairly straightforward corollaries of the Morita (uniqueness) theorem.
\begin{theorem} {\mathbf Corollary 1.}
Two rings, $A$ and $B$, are Morita equivalent if and only if there is an $(A,B)$-bimodule $M_{b}$ and a $(B,A)$-bimodule $N_{b}$ so that $M_{B}\bigotimesBN_{B}\simeq A$ as $A$-bimodules and $N_{B}\bigotimes_{A}M_{b}\simeq B$ as $B$-bimodules. With these assumptions, one obtains:
$End_{A-mod}(M_{b})=B^{op},$ $End_{B-mod}(N_{b})=A^{o}p$. Also $M_{b}$ is projective as an $A$-module, whereas $N_{B}$ is projective as a $B$-module. \end{theorem}
Proof . All equivalences of categories are exact functors, and therefore they preserve projective objects as required by Corollary 1.
\begin{theorem}Corollary 2.
* (i). If $A$ and $B$ are Morita equivalent rings, then the corresponding categories ${\mathbf {m} od-A}$ and ${\mathbf {m} od-B}$ are also equivalent.
* (ii). Furthermore, there exists a natural equivalence of categories ${\mathbf {A} -bimod}\to {\mathbf {B} -bimod}$ which takes $A$ to $B$, of course along with their natural bimodule structures.
\end{theorem}
Proof. Let $M_{b}$ and $N_{b}$ be the bimodules already defined in Corollary 1 .
For proposition (i), one utilizes the functors Failed to parse (syntax error): {\displaystyle (âˆ’ \bigotimes_A M_b} and Failed to parse (syntax error): {\displaystyle (âˆ’ \bigotimes_B N_b)} to prove the equivalence of the two categories.
For the second proposition (ii), one needs to employ the functor $N_{b}\bigotimes_{A}-\bigotimes_{A}M_{b}:{\mathbf {A} -bimod}\longrightarrow {\mathbf {B} -bimod}$ to prove the natural equivalence of the latter two categories.
# PlanetPhysics/Muon

Muon is a 1/2 spin fermion and it's one of the fundamental physics particles known as leptons. Muons have similar characterstics to electron except that their mass is about 200 times greater. Muons that are found on Earth are mostly product of a reaction between cosmic rays and matter in Earth's athmosphere. Even though muons have a very short lifetime (2.2 microseconds) we still see a relatively large flux of muons on Earth's surface, since muons travel at high velocities and thust time dialation effect allows them to travel far.
Muons are one of the biggest sources of background radiation on Earth's surface. This is precisely why many experiments are moved deep into Earth in order to eliminate the background noise of muons.
# PlanetPhysics/Moment of Inertia of a Circular Disk

Here we look at two cases for the moment of inertia of a homogeneous circular disk
(a) about its geometrical axis,
(b) about one of the elements of its lateral surface.
Let m be the mass, a the radius, l the thickness, and $\tau$ the density of the disk. Then choosing a circular ring for the element of mass we have
$dm=\tau \cdot l\cdot 2\pi r\cdot dr$
where $r$ is the radius of the ring and $dr$ its thickness.
\begin{figure} \includegraphics[scale=.6]{Fig87.eps} \vspace{20 pt} \end{figure}
Therfore the moment of inertia about the axis of the disk is
$I=2\pi l\tau \int _{0}^{a}r^{3}dr$ $I={\frac {\tau l\pi a^{4}}{2}}$ $I={\frac {ma^{2}}{2}}$
The moment of inertia about the element is obtained easily by the help of theorem II. Thus
$I'=I+ma^{2}$ $I'={\frac {3}{2}}ma^{2}$
It will be noticed that the thickness of the disk does not enter into the expressions for $I$ and $I'$ except through the mass of the disk. Therefore these expressions hold good whether the disk is thick enough to be called a cylinder or thin enough to be called a circular lamina.
# PlanetPhysics/Morita Equivalence Lemma for Arbitrary Algebras
## Morita equivalence lemma for arbitrary algebras

Let us consider first an example of Morita equivalence; thus, for an integer $n\geq 1$, let $Mat_{n}(A)$ be the algebra of $n\times n$-matrices with entries in an algebra $A$. The following is a typical example of Morita equivalence that involves noncommutative algebras.
\begin{theorem}{\mathbf Morita equivalence Lemma for arbitrary algebras}
For any algebra $A$ and any integer $n\geq 1$, the algebras $A$ and $Mat_{n}(A)$ are Morita equivalent. \end{theorem}
{\mathbf Important Notes:}
* Even if $A$ is a commutative algebra, the algebra $Mat_{n}(A)$ is of course not commutative for any $n>1$ because the matrix multiplication is generally non-commutative.
* In general, the algebra $A$ cannot be recovered from its corresponding abelian category $A$-mod. Therefore, in order for a concept in noncommutative geometry to have or retain an intrinsic meaning, such a concept must be Morita invariant that is, to remain within the same Morita equivalence class. This raises the important question: what properties of an algebra are Morita invariant ? The answer to this question is provided by the Uniqueness Morita Theorem.
# PlanetPhysics/Multigrade Operator

A multigrade operator $\Omega$ is a parametric operator with parameter $k$ in the set $\mathbb {N}$ of non-negative integers.
The application of a multigrade operator $\Omega$ to a finite sequence of operands $(x_{1},\ldots ,x_{k})$ is typically denoted with the parameter $k$ left tacit, as the appropriate application is implicit in the number of operands listed. Thus $\Omega (x_{1},\ldots ,x_{k})$ may be taken for $\Omega _{k}(x_{1},\ldots ,x_{k}).$
# PlanetPhysics/Momentum

In classical mechanics, momentum is the product of mass and velocity of a moving object. More precisely, the momentum of a particle or rigid body with mass $m$ and velocity $v$ is defined as $\mathbf {p} \equiv m\mathbf {v} .$ For a collection of $n$ particles (or rigid bodies) moving with different velocities, the total momentum $\mathbf {p}$ is defined as the vector sum of the momenta of the particles, $\mathbf {p} _{1},\mathbf {p} _{2},\ldots ,\mathbf {p} _{n}$: $\mathbf {p} \equiv \sum _{i=1}^{n}\mathbf {p} _{i}.$ The SI unit of momentum is kilogram metre per second (kg m/s).
Momentum plays an important role in Newton's second law, which in its most concise formulation reads $\mathbf {F} =\mathbf {\dot {p}} ,$ expressing that the total force exerted on an object equals the time derivative of its momentum. In the special case that there are no forces acting on the object, we get back Newton's first law: $\mathbf {p}$ (and therefore $\mathbf {v}$) is constant. More generally, if in a system consisting of a number of objects there are no external forces (i.e., the only forces present are those between the objects themselves), then Newton's third law implies that all the forces cancel each other, so that the total momentum is still constant.
This last observation leads to the idea of a conserved quantity . It turns out that the law of conservation of momentum is equivalent to the invariance of the physical laws under translations. This idea can be extended: Noether's theorem shows that every continuous symmetry of a physical law that can be formulated as an action principle leads to a conserved quantity. In this setting, momentum can be viewed as the conserved quantity corresponding to spatial translations. This notion of momentum can be generalised in such a way that it is also possible to speak of the momentum of, for example, electromagnetic waves.
Momentum plays an important role in the Hamiltonian formalism of classical mechanics, where mechanical systems are described in terms of generalised coordinates and generalised momenta. This, in turn, explains the appearance of the momentum operator in the Schr\"odinger equation in quantum mechanics.
A quantity related to momentum is impulse , which is defined as the force exerted on a moving object integrated over a time interval $[t_{0},t_{1}]$: $\mathbf {I} \equiv \int _{t_{0}}^{t_{1}}\mathbf {F} (t)\,\mathrm {d} t.$ By integrating the two sides of Newton's second law over time, we obtain the equation $\mathbf {I} =\mathbf {p} (t_{1})-\mathbf {p} (t_{0}).$ In words: the impulse is equal to the change in momentum.
# PlanetPhysics/Morita Equivalence
## Morita equivalence

This entry presents both the definition of Morita equivalent algebras and the Morita equivalence theorem, with a brief proof included. Let $A$ and $B$ be two associative, but not necessarily commutative, algebras. Such algebras $A$ and $B$ are called Morita equivalent , if there is an equivalence of categories between $A$-mod and $B$-mod.
\begin{theorem}{\mathbf Morita Equivalence Theorem} Commutative algebras $A$ and $B$ are Morita equivalent if and only if they are isomorphic.\end{theorem}
Proof . Following the above definition, isomorphic algebras are Morita equivalent. Let us assume that $A$ and $B$ are any two such Morita equivalent associative algebras. It follows then that $A-mod\sim B-mod$, and thus one also has that $Z(A-mod)\simeq Z(B-mod).$ If $A$ and $B$ are both commutative, then by the Associative Algebra Lemma one also has that $A=Z_{A}$ and $B=Z_{B}.$
# PlanetPhysics/Monomorphism

Definition 0.1
A morphism Failed to parse (unknown function "\Map"): {\displaystyle \Map fAB} is a monomorphism , if for any two morphisms Failed to parse (unknown function "\Map"): {\displaystyle \Map{g,h}CA} the equality $f\circ g=f\circ h$ implies $h=g$.
# QuantLib/FinancialVisualizationInstallation

For Linux you will need to first download the following subversion repositories
Please e-mail me at joe@gnacademy.org if you have any problems with these instructions

## Dependencies

For this to work you will need to make sure that your linux box has installed a number of packages. The packages that are needed for the mandriva installation are
(For AMD64, you will need to use. the lib64 versions).

## Installation

For the QuantLib libraries, you can install with the following commands
The QuantLib-SWIG libraries can also be installed in this way, but you will have to cd to QuantLib-SWIG/Python to install the package
To build the enthought packages

## Environment

You will need to set PYTHONPATH to point to the enthought libraries. An example line to do this is
export PYTHONPATH=/home/joe/enthought/src/lib

## Running

To run the demo go into the chinaresearch directory and type
python option.plotspace.py
or
python option.basket.py
# Qualitative research methods

Qualitative methods in sociological research refer to distinctive types of research activities: participant observation, intensive interviewing, and focus groups. The three qualitative designs differ but also share many similar characteristics that distinguish them from quantitative methods of research. Qualitative researchers begin their research with an exploratory research question (for many times there isn't sufficient data to formulate a structured and specific goal). And once started they make sure to pay attention to the social context in which social phenomena occur, human subjectivity and how they themselves can influence any situation.

## Participant observation

Participant observation is a method for gathering data that involves developing a relationship with people while they go about their daily, normal activities. It is a means for seeing the social world as the research subjects see it, in its totality, and for understanding subjects' interpretations of that world (Wolcott, 1995:66).
For more information, see Participant observation (Wikipedia)

## Intensive Interviewing

is a method in which the researcher seeks in-dept information from their interviewee's feelings, experiences, and perceptions.

## Focus Groups

is a method in where the researcher seeks to encourage discussion among participants about a certain topic of interest.
# Qualitative MO theory

Qualitative molecular orbital (MO) theory is a way of building up a simple picture of the molecular orbitals of a molecule using a few basic rules.
The ideas discussed in this section will allow you to make 'back-of-the-envelope' predictions and allow you to interpret the results from more sophisticated calculations.
* Atomic orbitals to molecular orbitals
* Molecular orbitals for H$_2$O and CH$_2$
* Symmetry tutorial
* MOs for methyl: CH$_3$., CH$_3$$^+$, CH$_3$$^-$
* Walsh diagrams
* Qualitative MO theory comprehension test
| | Computational Chemistry - Navigation | |
| Introduction to approximate quantum chemistry | Fundamentals of Computational Chemistry | Quantitative MO theory |
# Qualitative analysis

* Qualitative analysis involves systematic coding, organisation, and analysis of non-numerical data.
# Qualitative analysis/Data analysis tutorial

See Survey research and design in psychology/Tutorials/Qualitative analysis.
# Occupational Epidemiology/Research tools/Authors´s guidelines for scientific publications

By publication in PubMed indexed journals, the articles will be public searchable. Open access journal are preferred, to give access to all users and researchers globally, also in low income countries.

## Open access journals (free for users and publishers)

* International Maritime Health free to publish, open access and PubMed indexed
* Indian Journal of Public Health: low cost to publish, open access and PubMed indexed
* International Journal of Travel Medicine and Global Health no publication fees, open acces and PubMed indexed
* Medicina Maritima free to publish and to read - articles can be searched here and through Int Mar Health

## Authors´ guidelines

* International Maritime Health
* Indian Journal of Public Health
* International Journal of Travel Medicine and Global Health
* Medicina Maritime
PRISMA statement for the reporting of systematic reviews and meta-analyses, the STROBE statement for the reporting of epidemiological studies, and the RECORD statement for the reporting of studies conducted using observational routinely-collected health data. Authors should check the EQUATOR Network for reporting instructions and further information.
# Power Systems/Sources

Chemical (Batteries) Nuclear (fission & Fusion) Natural Resurces (Coal, Natural Gas, Steam, Water Driven, Solar)
# Power Systems/Synchronous Generator

synchronous generator are efficient form for generating electricity than conventional DC generator. Many industries started generating through synchronous generator for constant voltage at constant frequency. it is better than alternator in aspects like large power rating and efficient control.
# Power Systems/Ybus

Impedance Diagram
# Power function/Positive base/Real exponent/Fact/Proof

By definition, we have
    $x^{\alpha }=\exp {\left(\alpha \,\ln x\right)}\,.$
The derivative with respect to $x$ equals
    ${\left(x^{\alpha }\right)}'={\left(\exp {\left(\alpha \,\ln x\right)}\right)}'={\frac {\alpha }{x}}\cdot \exp {\left(\alpha \,\ln x\right)}={\frac {\alpha }{x}}x^{\alpha }=\alpha x^{\alpha -1}\,$
using fact, fact and the chain rule.
# Power function/R/Determine derivative/x^n/n in N/Exercise

Determine the derivative of the functions
    $\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto x^{n},$
for all $n\in \mathbb {N}$.
# Power Systems/Introduction

Power Systems is the study of generating, transmitting and distributing electricity from power generating facilities to customer ends.
# Power function/Positive base/Real exponent/Continuous/Exercise

Let $u\in \mathbb {R}$. Show that the power function
    $f\colon \mathbb {R} _{+}\longrightarrow \mathbb {R} ,x\longmapsto x^{u},$
is continuous.
# Power function/Positive base/Real exponent/Fact

Let $\alpha \in \mathbb {R}$.
function
    $f\colon \mathbb {R} _{+}\longrightarrow \mathbb {R} _{+},x\longmapsto x^{\alpha },$
is differentiable, and its derivative is
    $f'(x)=\alpha x^{\alpha -1}\,.$
# Power series/R/Fourth power/Fifth coefficient/Exercise

    $\sum _{n=0}^{\infty }a_{n}x^{n}$
$x^{0},x^{1},x^{2},x^{3},x^{4},x^{5}$
    $\sum _{n=0}^{\infty }c_{n}x^{n}={\left(\sum _{n=0}^{\infty }a_{n}x^{n}\right)}^{4}\,.$
# Real numbers/Series/Cauchy-criterion/Fact/Proof/Exercise/Solution

We set $x_{n}:=\sum _{k=0}^{n}a_{k}$. The convergence of the series means the convergence of this sequence of partial sums. A real sequence converges if and only if it is a Cauchy-sequence. This means that for every $\epsilon >0$ there exists some $n_{0}$ such that for all
    $n\geq m\geq n_{0}\,$
the estimate
    $\vert {x_{n}-x_{m}}\vert \leq \epsilon \,$
holds. In the case of a series this just means
    $\vert {x_{n}-x_{m}}\vert =\vert {\sum _{k=0}^{n}a_{k}-\sum _{k=0}^{m}a_{k}}\vert =\vert {\sum _{k=m+1}^{n}a_{k}}\vert \leq \epsilon \,.$
# Real numbers/Several subsets/Concepts of boundedness/Exercise

Examine for each of the following subsets $M\subseteq \mathbb {R}$
* $\{2,-3,-4,5,6,-1,1\}$,
* $\left\{{\frac {1}{2}},{\frac {-3}{7}},{\frac {-4}{9}},{\frac {5}{9}},{\frac {6}{13}},{\frac {-1}{3}},{\frac {1}{4}}\right\}$,
* $]-5,2]$,
* ${\left\{{\frac {1}{n}}\mid n\in \mathbb {N} _{+}\right\}}$,
* ${\left\{{\frac {1}{n}}\mid n\in \mathbb {N} _{+}\right\}}\cup \{0\}$,
* $\mathbb {Q} _{-}$,
* ${\left\{x\in \mathbb {Q} \mid x^{2}\leq 2\right\}}$,
* ${\left\{x\in \mathbb {Q} \mid x^{2}\leq 4\right\}}$,
* ${\left\{x^{2}\mid x\in \mathbb {Z} \right\}}$.
# Real numbers/Subset/Supremum/Definition

For a nonempty subset $M\subseteq \mathbb {R}$, an upper bound $T$ of $M$ is called the supremum of $M$, if $T\leq S$ holds for all upper bounds $S$ of $M$.
# Real numbers/Series/Rules/Fact/Proof/Exercise

Let
    $\sum _{k=0}^{\infty }a_{k}{\text{ and }}\sum _{k=0}^{\infty }b_{k}$
be convergent series of real numbers with sums $s$ and $t$. Prove the following statements.
* The series $\sum _{k=0}^{\infty }c_{k}$ with $c_{k}=a_{k}+b_{k}$ is convergent with sum equal to $s+t$.
* For $r\in \mathbb {R}$ the series $\sum _{k=0}^{\infty }d_{k}$ with $d_{k}=ra_{k}$ is also convergent with sum equal to $rs$.
# Real numbers/Series converges/Summands/Null sequence/Fact

Let
    $\sum _{k=0}^{\infty }a_{k}$
denote a convergent series of real numbers.
    $\lim _{k\rightarrow \infty }a_{k}=0\,.$
# Real numbers/Subsequence/Definition

Let ${\left(x_{n}\right)}_{n\in \mathbb {N} }$ be a real sequence. For any strictly increasing mapping $\mathbb {N} \rightarrow \mathbb {N} ,i\mapsto n_{i}$, the sequence
    $i\mapsto x_{n_{i}}$
is called a subsequence of the sequence.
# Real numbers/Series/Rules/Fact

Let
    $\sum _{k=0}^{\infty }a_{k}{\text{ and }}\sum _{k=0}^{\infty }b_{k}$
denote convergent series of real numbers with sums $s$ and $t$ respectively. Then the following statements hold.
* The series $\sum _{k=0}^{\infty }c_{k}$ given by $c_{k}:=a_{k}+b_{k}$ is also convergent and its sum is $s+t$.
* For $r\in \mathbb {R}$ also the series $\sum _{k=0}^{\infty }d_{k}$ given by $d_{k}:=ra_{k}$ is convergent and its sum is $rs$.
# Real numbers/Series converges/Summands/Null sequence/Fact/Proof

This follows directly from fact.
# Real numbers/Subset/Infimum/Definition

For a nonempty subset $M\subseteq \mathbb {R}$, a lower bound $t$ of $M$ is called the infimum of $M$, if $t\geq s$ holds for all lower bounds $s$ of $M$.
# Primitive functions/Relation to fundamental theorem/Section
## Definition

Let $I\subseteq \mathbb {R}$ denote an interval, and let
    $f\colon I\longrightarrow \mathbb {R}$
denote a function. A function
    $F\colon I\longrightarrow \mathbb {R}$
is called a primitive function for $f$, if $F$ is differentiable on $I$ and if $F'(x)=f(x)$ holds for all
$x\in I$
A primitive function is also called an antiderivative. The fundamental theorem of calculus might be rephrased, in connection with fact, as an existence theorem for primitive functions.

## Corollary

Let $I$ denote a real interval, and let
    $f\colon I\longrightarrow \mathbb {R}$
denote a continuous function. Then $f$ has a

### Proof

Let $a\in I$ be an arbitrary point. Due to fact, there exists the function
    $F(x)=\int _{a}^{x}f(t)\,dt\,,$
and because of the Fundamental theorem, the identity $F'(x)=f(x)$ holds. This means that $F$ is a primitive function for $f$.
$\Box$

## Lemma

Let $I$ denote a real interval, and let
    $f\colon I\longrightarrow \mathbb {R}$
denote a function. Suppose that $F$ and $G$ are primitive functions of $f$. Then $F-G$ is a

### Proof

We have
    $(F-G)'=F'-G'=f-f=0\,.$
Therefore, due to fact, the difference $F-G$ is constant.
$\Box$
The following statement is also a version of the fundamental theorem, it is called the Newton-Leibniz-formula.

## Corollary

Let $I$ denote a real interval, and let
    $f\colon I\longrightarrow \mathbb {R}$
denote a continuous function. Suppose that $F$ is a primitive function for $f$. Then for $a,b\in I$, the identity
    $\int _{a}^{b}f(t)\,dt=F(b)-F(a)\,$

### Proof

Due to fact, the integral exists. With the integral function
    $G(x):=\int _{a}^{x}f(t)\,dt\,,$
we have the relation
    $\int _{a}^{b}f(t)\,dt=G(b)=G(b)-G(a)\,.$
Because of fact, the function $G$ is differentiable and
    $G'(x)=f(x)\,$
holds. Hence $G$ is a primitive function for $f$. Due to fact, we have $F(x)=G(x)+c$. Therefore,
    $\int _{a}^{b}f(t)\,dt=G(b)-G(a)=F(b)-c-F(a)+c=F(b)-F(a)\,.$
$\Box$
Since a primitive function is only determined up to an additive constant, we sometimes write
    $\int _^f(t)\,dt=F+c\,.$
Here $c$ is called a constant of integration. In certain situations, in particular in relation with differential equations, this constant is determined by further conditions.