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11-07-2025 - Mathematical Analysis - Integrals of rational functions [EN]-[IT] by stefano.massari

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hive-146620 · @stefano.massari ·
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11-07-2025 - Mathematical Analysis - Integrals of rational functions [EN]-[IT]
![image.png](https://files.peakd.com/file/peakd-hive/stefano.massari/EpA4umEh1RRVH7PiX1S12agUt2prxiWfrLtXbxqTCxuZFyRteMcTkK7oRtiPy3GoZ6q.png)


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*~~~ La versione in italiano inizia subito dopo la versione in inglese ~~~*

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**ENGLISH**
https://images.ecency.com/DQmNuqRpdWgTaWvjwbmaVWSemm13V7viV9jyRyVFHiMSbYA/optimized_image_1_.jpeg 

**11-07-2025 - Mathematical Analysis - Integrals of rational functions [EN]-[IT]**
With this post I would like to give a short instruction about the topic mentioned in the subject
(code notes: X-60-59)

***Integrals of rational functions***

![image.png](https://files.peakd.com/file/peakd-hive/stefano.massari/AJkRqytDuBuGx2H2J1GS6e1XGpp6tmHgQHtWLZMu5qR3y9ewG2rYFX6nB2S86Kh.png)

*image created with artificial intelligence, the software used is Microsoft Copilot*

**Introduction**
Let's start with a warning, the integration of rational functions can be a difficult topic to digest.
First of all, let's talk about rational function. A rational function is obtained as a ratio of two polynomials with real coefficients. Example below.

![image.png](https://files.peakd.com/file/peakd-hive/stefano.massari/23twAKVLLereMv7Tc9cbfwcE6rxigXVDZRLaWHD6wi23o5VYA8iwT4twdVJVUJXwJZDy1.png)
NOTE: a polynomial is an algebraic expression formed by a sum of monomial terms of the following type:

![image.png](https://files.peakd.com/file/peakd-hive/stefano.massari/23viTbaMod1NJuHFkmjFgeQc4fjQ4iwhxbQtzyK8EiZD6MRqNKNagYFjwUvoKKRnNausi.png)

Where:
ak= coefficient belonging to a given numerical set
x = variable
k = a non-negative integer

The general form of a polynomial is as follows:

![image.png](https://files.peakd.com/file/peakd-hive/stefano.massari/23w3EneigMm78E4PqVbvGVa2YpEW23fXF9QX8NwP3wwAfEzTh6LUmERf2vggzHQM1fvof.png)

*example*
Here is an example of a polynomial of degree 4, principal coefficient 3, known term -5

![image.png](https://files.peakd.com/file/peakd-hive/stefano.massari/23xLMe4GPNzo36nLBqmwNpbxm5EgaFtmfvDSKNGB9rfLmJFhDbbR5oAQAzUPgA6LmEtH3.png)

**characteristics of a real function**
-Domain D, i.e. all real numbers do not cancel the denominator
-As a function it has the characteristic of being continuous.
-Derivatives are rational

Here is the appearance of a rational function:

![image.png](https://files.peakd.com/file/peakd-hive/stefano.massari/23tmmjqTG8n2zDxStaH4zkJSLCHDXDfYxh5rWeZ7o8TFJhqjkyJExcnGjfaFJtDuEAFvp.png)

**Integrations of rational functions**
We have tried to transfer the concept of a rational function which in summary we can say is a ratio of polynomials.
The integral of a rational function is calculated with a procedure that is divided into three steps. These steps are called as follows:
-Division
-Decomposition into simple fractions
-Integration of each term

*Exercise*
If F(x) is the primitive of 2(2x2+x)/(2x-1) which is 3 in 1, then what is F(2)?
This exercise is certainly complicated, but I will try to show the steps to do it anyway.

Let's start from its description:

![image.png](https://files.peakd.com/file/peakd-hive/stefano.massari/23uFwixBoLn4pFoG2TznDKdkN9pBdzXTnAHzDURnNd1K8T99p148PVKxz4fY3gVou9PZa.png)

let's apply the first transformation

![image.png](https://files.peakd.com/file/peakd-hive/stefano.massari/23uRKjMYCZ2W8jzerMbbKxpfTfNCGm4LNDHEJRS6HPm3sw854ZGUfbbu9syQondhxyqyH.png)

then we arrive at the form in which the constant is present C.

![image.png](https://files.peakd.com/file/peakd-hive/stefano.massari/23uRKjMsNiiREMP24aDw4Lb43ekSyB3P5NqHqhAPshsg2faLxKkzsg3t4BRdFkcE1CJtf.png)

After a short reasoning we notice that F(1) = 3 gives the following condition:

![image.png](https://files.peakd.com/file/peakd-hive/stefano.massari/23uRKjMCzmUE6qTSPs57YA8xV6qzFZPa1qbLHGHixR2jxPwnhPT4dd5Z8DZtX6t9DEfNP.png)

So we can write F(x)

![image.png](https://files.peakd.com/file/peakd-hive/stefano.massari/23viTgJPR9XZdDuZzr4jwNcVFNzqHAafu9Fpy32gWADmmZyCtaeaM58s7wJxHEWdpNosZ.png)

And now we calculate the function at x=2

![image.png](https://files.peakd.com/file/peakd-hive/stefano.massari/23uRLPLAqtvyjPHoZhRUVSm4yzv6srStD9kvPWi4t8wvZ5NUuAGgEkCkiqY4huc7N6nwJ.png)

*Result*
the result is **8+ln3**
![image.png](https://files.peakd.com/file/peakd-hive/stefano.massari/23wN4BmfPx9wQA93fHE2r6LiYY6S9U1K9Y7MmbyD2mNstpkhMtWR6mxUPhd7kUKaHYmej.png)

***Conclusions***
Integrating rational functions is useful in analytic geometry, physics, electrical engineering, and statistics.

***Question***
Did you know that Leonhard Euler, a Swiss mathematician and astronomer (also known as Euler) provided the practical tools for integrating rational functions that students use every day today?





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**ITALIAN**
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**11-07-2025 - Analisi Matematica - Integrali di funzioni razionali [EN]-[IT]**
Con questo post vorrei dare una breve istruzione a riguardo dell’argomento citato in oggetto
(code notes: X-60-59)

***Integrali di funzioni razionali***



![image.png](https://files.peakd.com/file/peakd-hive/stefano.massari/AJkRqytDuBuGx2H2J1GS6e1XGpp6tmHgQHtWLZMu5qR3y9ewG2rYFX6nB2S86Kh.png)


*immagine creata con l’intelligenza artificiale, il software usato è Microsoft Copilot*

**Introduzione**
Partiamo subito con un avviso, l'integrazione di funzioni razionali può essere un argomento difficile da digerire.
Innanzitutto parliamo di funzione razionale. Una funzione razionale è ottenuta come rapporto di due polinomi a coefficienti reali. Esempio qui sotto.


![image.png](https://files.peakd.com/file/peakd-hive/stefano.massari/23twAKVLLereMv7Tc9cbfwcE6rxigXVDZRLaWHD6wi23o5VYA8iwT4twdVJVUJXwJZDy1.png)
NOTA: un polinomio è un’espressione algebrica formata da una somma di termini monomî del tipo seguente:

![image.png](https://files.peakd.com/file/peakd-hive/stefano.massari/23viTbaMod1NJuHFkmjFgeQc4fjQ4iwhxbQtzyK8EiZD6MRqNKNagYFjwUvoKKRnNausi.png)

Dove:
ak= coefficiente appartenente a un dato insieme numerico
x = variabile
k = un numero intero non negativo

La forma generale di un polinomio è la seguente:


![image.png](https://files.peakd.com/file/peakd-hive/stefano.massari/23w3EneigMm78E4PqVbvGVa2YpEW23fXF9QX8NwP3wwAfEzTh6LUmERf2vggzHQM1fvof.png)

*esempio*
Qui di seguito l'esempio di un polinomio di grado 4, coefficiente principale 3, termine noto -5


![image.png](https://files.peakd.com/file/peakd-hive/stefano.massari/23xLMe4GPNzo36nLBqmwNpbxm5EgaFtmfvDSKNGB9rfLmJFhDbbR5oAQAzUPgA6LmEtH3.png)




**caratteristiche di una funzione reale**
-Dominio D, cioè tutti i numeri reali non annullano il denominatore
-Come funzione ha la caratteristica di essere continua.
-Le derivate risultano razionali

Qui di seguito l'aspetto di una funzione razionale:

![image.png](https://files.peakd.com/file/peakd-hive/stefano.massari/23tmmjqTG8n2zDxStaH4zkJSLCHDXDfYxh5rWeZ7o8TFJhqjkyJExcnGjfaFJtDuEAFvp.png)

**Integrazioni di funzioni razionali**
Abbiamo cercato di trasferire il concetto di una funzione razionale che in sintesi  possiamo dire che è un rapporto di polinomi.
L'integrale di una funzione razionale si calcola con un procedimento che è diviso in tre passi. Questi passi sono chiamati come segue:
-Divisione
-Decomposizione in fratti semplici
-Integrazione di ogni termine

*Esercizio*
Se F(x) è la primitiva di 2(2x2+x)/(2x-1) che vale 3 in 1, allora F(2) quanto vale?
Questo esercizio è sicuramente complicato, ma provo comunque a mostrare i passi per svolgerlo.

Partiamo dalla sue descriizione:

![image.png](https://files.peakd.com/file/peakd-hive/stefano.massari/23uFwixBoLn4pFoG2TznDKdkN9pBdzXTnAHzDURnNd1K8T99p148PVKxz4fY3gVou9PZa.png)

applichiamo la prima trasformazione

![image.png](https://files.peakd.com/file/peakd-hive/stefano.massari/23uRKjMYCZ2W8jzerMbbKxpfTfNCGm4LNDHEJRS6HPm3sw854ZGUfbbu9syQondhxyqyH.png)

poi arriviamo alla forma in cui è presente la costante C.

![image.png](https://files.peakd.com/file/peakd-hive/stefano.massari/23uRKjMsNiiREMP24aDw4Lb43ekSyB3P5NqHqhAPshsg2faLxKkzsg3t4BRdFkcE1CJtf.png)

Dopo un breve ragionamento notiamo che F(1) = 3 fornisce la seguente condizione:


![image.png](https://files.peakd.com/file/peakd-hive/stefano.massari/23uRKjMCzmUE6qTSPs57YA8xV6qzFZPa1qbLHGHixR2jxPwnhPT4dd5Z8DZtX6t9DEfNP.png)


Quindi possiamo scrivere F(x)


![image.png](https://files.peakd.com/file/peakd-hive/stefano.massari/23viTgJPR9XZdDuZzr4jwNcVFNzqHAafu9Fpy32gWADmmZyCtaeaM58s7wJxHEWdpNosZ.png)


Ed ora calcoliamo la funzione in x=2


![image.png](https://files.peakd.com/file/peakd-hive/stefano.massari/23uRLPLAqtvyjPHoZhRUVSm4yzv6srStD9kvPWi4t8wvZ5NUuAGgEkCkiqY4huc7N6nwJ.png)

*Risultato*
il risultato è **8+ln3**
![image.png](https://files.peakd.com/file/peakd-hive/stefano.massari/23wN4BmfPx9wQA93fHE2r6LiYY6S9U1K9Y7MmbyD2mNstpkhMtWR6mxUPhd7kUKaHYmej.png)


***Conclusioni***
L’integrazione di funzioni razionali è utile nell'ambito della geometria analitica, in fisica, in elettrotecnica e nella statistica.

***Domanda***
Lo sapevate che Leonhard Euler, matematico e astronomo svizzero (conosciuto anche come Eulero) fornì gli strumenti pratici per integrare le funzioni razionali che gli studenti usano oggi quotidianamente?

**THE END**
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@bisolamih ·
$0.10
This is entirely new to me
It’s an amazing topic anyway
Nice one!
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@stefano.massari · 
Integrating rational functions is, in my opinion, a complex topic. It all starts with understanding how to write the integral, which I've included below. The integral is represented almost everywhere with an elongated letter s, which describes the interval of integration; in the example below, it's written in blue. After the integral, the function to be integrated is described; in the example below, it's written in red. Finally, we need to describe where we add the slices of infinitesimal width, and in this case, we do so on the x-axis (in the example below, the term representing this is written in black).
![](https://images.ecency.com/DQmczHQwKcXGDvvJNVoqbmaTw22V8gpetMuz764D2R2w8RK/image.png)
!CTP
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@commentrewarder · (edited)
$0.06
@stefano.massari, I paid out 0.166 HIVE and 0.035 HBD to reward 10 comments in this discussion thread.
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@davideownzall · 
Polinomi e monomi, altro ricordo sbloccato... Questo esercizio effettivamente non era proprio banale, ora da solo senza la tua guida non sarei riuscito a farlo 🤣

!PIZZA
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@stefano.massari · 
Coi polinomi ricordo di aver fatto fatica... in realtà non sono complessi, ma la parola mi incuteva timore...hahah e forse mi irrigidivo durante la spiegazione e non la comprendevo. La spiegazione più semplice è dire che un polinomio è una somma di monomi. E qui se uno non sa che cosa è un monomio è fregato. Comunque un monomi è un termino di questo tipo:
![](https://images.ecency.com/DQmVzLc1J6KTWqKqxsNsaARXaHcWjM74wgsoV5B6ykAXDVz/image.png)

Dove:
a = coefficiente
x = variabile
n = esponente intero non negativo
Un operazione di base con un polinomio può essere la seguente:
![](https://images.ecency.com/DQmVSHNhGY5iTaRhFZurF5Xu9HgdA6ur8ZvSC5GmHRhX6wf/image.png)

!WINE
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@davideownzall ·
$0.11
Si il monomio era facile alla fine della fiera 🤣

!PIZZA
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@discovery-it ·
$0.02
<div class="pull-left">https://cdn.steemitimages.com/DQmTAn3c753LR7bHCLPo96g9UvRMaPFwaMYn8VQZa85xczC/discovery_logo_colore%20-%20Copia.png</div><br> This post was shared and voted inside the discord by the curators team of <a href="https://discord.gg/cMMp943"> discovery-it</a> <br>Join our <a href = "https://hive.blog/trending/hive-193212"> Community</a> and follow our <a href = "https://hive.vote/dash.php?i=1&trail=discovery-it">Curation Trail</a><br>Discovery-it is also a Witness, vote for us <a href = "https://hivesigner.com/sign/account-witness-vote?witness=discovery-it&approve=true"> here</a>  <br>Delegate to us for passive income. Check our <a href = "https://hive.blog/hive-193212/@discovery-it/delegations-program-80-fee-back"> 80% fee-back Program</a> <hr>
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@goshen ·
$0.11
Thank you so much for taking quality time of yours to actually explain this in deep. I really appreciate
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@stefano.massari · 
Thank you for your kind words. To fully understand these concepts, it's essential to have a good understanding of integrals. For example, let's try to calculate a simple integral. Let's calculate the integral of x^2 in the interval from 0 to 2, which mathematically is written as follows:
![](https://images.ecency.com/DQmY4F9dmTjNNAqsDWhXSgU6nL8Kr5986JMj879sojbFUou/image.png)
Now we find the primitive which is the following
![](https://images.ecency.com/DQmQemj4cck4KuN5YtwbqUTDCAyb86xkNBbewurWiuQMrdx/image.png)
and we apply the fundamental theorem of calculus
![](https://images.ecency.com/DQmRehVfa5NUTumMkGXkfM8GcYVQndCirWDbHLxbe2iofT7/image.png)
The numerical result is 8/3
!BBH
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@hivebuzz · 
Congratulations @stefano.massari! You have completed the following achievement on the Hive blockchain And have been rewarded with New badge(s)

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@libertycrypto27 ·
$0.11
!discovery 30
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@beerlover · 
<div class='pull-right'>https://files.peakd.com/file/peakd-hive/beerlover/yiuU6bdf-beerlover20gives20BEER.gif<p><sup><a href='https://hive-engine.com/?p=market&t=BEER'>View or trade </a> <code>BEER</code>.</sup></p></div><center><br> <p>Hey @libertycrypto27, here is a little bit of <code>BEER</code> from @stefano.massari for you. Enjoy it!</p> <p>Learn how to <a href='https://peakd.com/beer/@beerlover/what-is-proof-of-stake-with-beer'>earn <b>FREE BEER</b> each day </a> by staking your <code>BEER</code>.</p> </center><div></div>
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@beerlover · 
<div class='pull-right'>https://files.peakd.com/file/peakd-hive/beerlover/yiuU6bdf-beerlover20gives20BEER.gif<p><sup><a href='https://hive-engine.com/?p=market&t=BEER'>View or trade </a> <code>BEER</code>.</sup></p></div><center><br> <p>Hey @libertycrypto27, here is a little bit of <code>BEER</code> from @stefano.massari for you. Enjoy it!</p> <p>Learn how to <a href='https://peakd.com/beer/@beerlover/what-is-proof-of-stake-with-beer'>earn <b>FREE BEER</b> each day </a> by staking your <code>BEER</code>.</p> </center><div></div>
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@stefano.massari · 
Grazie Liberty per il tuo prezioso sostegno. Ogni mese faccio sempre dei passi in avanti con HIVE. Grazie soprattutto al tuo supporto. Ho passato i 20.000 HP qualche settimana fa e adesso sono molto contento di come stanno andando le cose. Quando avevo iniziato a fare questa tipologia di contenuti tecnici pensavo che prima o poi avrei dovuto smettere, invece, grazie soprattutto al tuo supporto riesco a continuare a farli e ad avere delle soddisfazioni. !BEER
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@lupega ·
$0.12
Si potrebbe dire che quest'uomo,  è ancora attuale poiché gli studenti mettono in pratica ciò che ha lasciato come eredità.
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@stefano.massari · 
Leonhard Euler, che in Italia conosciamo come Eulero, ha  dato contributi fondamentali in matematica, fisica, ingegneria, astronomia. Non so se ricordi, ma di lui esistono anche oggi tante cose in matematica che portano il suo nome:
**La formula di Eulero**
![](https://images.ecency.com/DQmdeW3r1A4xhm9W1Xpi5MZqhgQ1LQoXm6a7CSNaom2DARU/image.png)
**Il numero di Eulero** (che oggi è la base dei logaritmi naturali)
![](https://images.ecency.com/DQmai2ZdsRCQZGRYhA7R6X8q7AfAp6nKKecsiSfW9Y4Neo1/image.png)
e una delle formule che alcuni matematici ritengono tra le più belle mai scritte, la formula **Identità di Eulero**:
![](https://images.ecency.com/DQma75oS8FtMTrdvqKHNixv3C54KWcwYFmeG24paTGkzfed/image.png)
!DIY
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@pizzabot · (edited)
RE: 11-07-2025 - Mathematical Analysis - Integrals of rational functions [EN]-[IT]
<center>PIZZA!


$PIZZA slices delivered:
@davideownzall<sub>(5/15)</sub> tipped @stefano.massari (x2)


<sub>Come get [MOON](https://moon.hive.pizza)ed!</sub></center>
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@siva1993 · 
Very good calculation
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@stefano.massari · 
In this article I also talked about primitives. In mathematical analysis, finding the primitive of a function means determining what we should derive to obtain that function. Below is an example where f(x) is on the left, and F(x) is on the right.
![](https://images.ecency.com/DQmXyo5oswu6bBWikJZBUtSMMmoSPy11KT2QHfMaxUrPhgm/image.png)
!ALIVE
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@siva1993 · 
great, in My school i learned as SinX / CosX = TanX
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