Apprends les Primitives et Dérivées Facilement avec PDF et Exos Corrigés

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Gabriel Cortes

19/04/2023

Maths

Équations différentielles + Intégrales

Matières

Maths

Apprends les Primitives et Dérivées Facilement avec PDF et Exos Corrigés

Name: Knowunity
Brand: Knowunity

Voici le résumé optimisé en français :

Les primitives des fonctions trigonométriques et les équations différentielles sont des concepts mathématiques fondamentaux. Ce document explore leurs propriétés, solutions et applications, en mettant l'accent sur les fonctions trigonométriques, les équations différentielles et les intégrales. Il aborde également des techniques avancées comme l'intégration par parties et les propriétés des intégrales définies.

Points clés :
• Relation entre dérivées et primitives des fonctions trigonométriques
• Solutions des équations différentielles de type y' = ay et y' = ay + f
• Concept d'intégrale comme aire sous la courbe
• Propriétés et techniques d'intégration
• Formules trigonométriques importantes

...

19/04/2023

1676

Derives et primitives des fonctions trigonométriques
los.
sin
- Sin
K
primitive
derivee
Cos
Demonstration: Deux primitives d'une même foncti

Voir

Differential Equations and Integration

This page delves deeper into differential equations and introduces the concept of integration.

The page begins by discussing differential equations of the form y' = ay + f, where f is a function.

Definition: The solutions to y' = ay + f are functions of the form y(x) = u(x) + v(x), where u(x) is a particular solution to y' = ay + f, and v(x) is any solution to y' = ay.

A special case is then presented: y' = ay + b, where b is a constant. The particular solutions for this case are constant functions of the form y_p(x) = -b/a.

The page then transitions to the topic of integrals. It provides a geometric interpretation of definite integrals as the area between the curve of a function and the x-axis over a given interval.

Highlight: The fundamental theorem of calculus is introduced: If F(x) = ∫_a^x f(t)dt, then F'(x) = f(x).

Several important properties of integrals are listed, including:

If f is negative, its integral will also be negative.
If f is both positive and negative over the interval, the areas should be calculated separately and then added.

The page concludes with a reminder about the trapezoidal rule for approximating integrals.

Example: The area of a trapezoid is given by (small base + large base) × height / 2.

This page provides essential information for students learning about differential equations and integration, bridging the gap between these two fundamental areas of calculus.

Voir

Advanced Integration Techniques and Properties

This page focuses on advanced integration techniques and properties, particularly the method of integration by parts.

The page begins with a detailed derivation of the fundamental theorem of calculus, showing how the derivative of an integral function equals the original function.

Highlight: The fundamental theorem of calculus states that if F(x) = ∫_a^x f(t)dt, then F'(x) = f(x).

The page then introduces the concept of integration by parts, a powerful technique for integrating products of functions.

Definition: Integration by parts formula: ∫u(x)v'(x)dx = [u(x)v(x)] - ∫u'(x)v(x)dx

An example is provided to illustrate how to apply this technique:

Example: To integrate x^2 cos(x), we can let u = x^2 and v' = cos(x). This transforms the integral into a more manageable form.

The page also mentions an important property of continuous functions:

Vocabulary: Every continuous function on an interval of R has primitives on that interval, although we may not always be able to find them explicitly (e.g., e^(-x^2)).

This information is crucial for students learning advanced integration techniques and understanding the limitations of analytical methods in calculus.

Voir

Properties of Integrals and Trigonometric Identities

This final page provides a comprehensive list of properties of integrals and important trigonometric identities.

The page begins with several key properties of definite integrals:

Highlight:

∫_a^b kf(x)dx = k∫_a^b f(x)dx (linearity)
∫_a^b [f(x) + g(x)]dx = ∫_a^b f(x)dx + ∫_a^b g(x)dx (additivity)
∫_a^c f(x)dx = ∫_a^b f(x)dx + ∫_b^c f(x)dx (additivity over intervals)

Additional properties are listed, including inequalities involving integrals and the concept of average value of a function.

Definition: The average value of a function f on [a,b] is given by (1/(b-a)) ∫_a^b f(x)dx.

The page concludes with a list of important trigonometric identities:

Vocabulary:

cos(a+b) = cos(a)cos(b) - sin(a)sin(b)
cos(a-b) = cos(a)cos(b) + sin(a)sin(b)
sin(a+b) = sin(a)cos(b) + cos(a)sin(b)
sin(a-b) = sin(a)cos(b) - cos(a)sin(b)
cos(2a) = cos^2(a) - sin^2(a) = 1 - 2sin^2(a) = 2cos^2(a) - 1
sin(2a) = 2cos(a)sin(a)

These identities are crucial for solving complex trigonometric problems and are often used in calculus and physics applications.

This page serves as a valuable reference for students working with integrals and trigonometric functions, providing a concise summary of key properties and identities.

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Knowunity est la meilleure application scolaire dans cinq pays européens.

Knowunity a été mis en avant par Apple et a toujours été en tête des classements de l'App Store dans la catégorie Éducation en Allemagne, en Italie, en Pologne, en Suisse et au Royaume-Uni. Rejoins Knowunity aujourd'hui et aide des millions d'étudiants à travers le monde.

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Knowunity est la meilleure application scolaire dans cinq pays européens.

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Les élèsves utilisent Knowunity

#1

Dans les palmarès des applications scolaires de 17 pays

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Louis B., utilisateur iOS

J'aime tellement cette application [...] Je recommande Knowunity à tout le monde ! !! Je suis passé de 11 à 16 grâce à elle :D

Stefan S., utilisateur iOS

L'application est très simple à utiliser et bien faite. Jusqu'à présent, j'ai trouvé tout ce que je cherchais :D

Lola, utilisatrice iOS

J'adore cette application ❤️ Je l'utilise presque tout le temps pour réviser.

Matières

Maths

Apprends les Primitives et Dérivées Facilement avec PDF et Exos Corrigés

Gabriel Cortes

@gab_cts

68 Abonnés

Suivre

Voici le résumé optimisé en français :

...

19/04/2023

1676

Tle

Maths

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Differential Equations and Integration

This page delves deeper into differential equations and introduces the concept of integration.

The page begins by discussing differential equations of the form y' = ay + f, where f is a function.

Definition: The solutions to y' = ay + f are functions of the form y(x) = u(x) + v(x), where u(x) is a particular solution to y' = ay + f, and v(x) is any solution to y' = ay.

A special case is then presented: y' = ay + b, where b is a constant. The particular solutions for this case are constant functions of the form y_p(x) = -b/a.

The page then transitions to the topic of integrals. It provides a geometric interpretation of definite integrals as the area between the curve of a function and the x-axis over a given interval.

Highlight: The fundamental theorem of calculus is introduced: If F(x) = ∫_a^x f(t)dt, then F'(x) = f(x).

Several important properties of integrals are listed, including:

If f is negative, its integral will also be negative.
If f is both positive and negative over the interval, the areas should be calculated separately and then added.

The page concludes with a reminder about the trapezoidal rule for approximating integrals.

Example: The area of a trapezoid is given by (small base + large base) × height / 2.

This page provides essential information for students learning about differential equations and integration, bridging the gap between these two fundamental areas of calculus.

Inscris-toi pour voir le contenu. C'est gratuit!

Accès à tous les documents

Améliore tes notes

Rejoins des millions d'étudiants

En t'inscrivant, tu acceptes les Conditions d'utilisation et la Politique de confidentialité.

Advanced Integration Techniques and Properties

This page focuses on advanced integration techniques and properties, particularly the method of integration by parts.

The page begins with a detailed derivation of the fundamental theorem of calculus, showing how the derivative of an integral function equals the original function.

Highlight: The fundamental theorem of calculus states that if F(x) = ∫_a^x f(t)dt, then F'(x) = f(x).

The page then introduces the concept of integration by parts, a powerful technique for integrating products of functions.

Definition: Integration by parts formula: ∫u(x)v'(x)dx = [u(x)v(x)] - ∫u'(x)v(x)dx

An example is provided to illustrate how to apply this technique:

Example: To integrate x^2 cos(x), we can let u = x^2 and v' = cos(x). This transforms the integral into a more manageable form.

The page also mentions an important property of continuous functions:

Vocabulary: Every continuous function on an interval of R has primitives on that interval, although we may not always be able to find them explicitly (e.g., e^(-x^2)).

This information is crucial for students learning advanced integration techniques and understanding the limitations of analytical methods in calculus.

Inscris-toi pour voir le contenu. C'est gratuit!

Accès à tous les documents

Améliore tes notes

Rejoins des millions d'étudiants

En t'inscrivant, tu acceptes les Conditions d'utilisation et la Politique de confidentialité.

Properties of Integrals and Trigonometric Identities

This final page provides a comprehensive list of properties of integrals and important trigonometric identities.

The page begins with several key properties of definite integrals:

Highlight:

∫_a^b kf(x)dx = k∫_a^b f(x)dx (linearity)
∫_a^b [f(x) + g(x)]dx = ∫_a^b f(x)dx + ∫_a^b g(x)dx (additivity)
∫_a^c f(x)dx = ∫_a^b f(x)dx + ∫_b^c f(x)dx (additivity over intervals)

Additional properties are listed, including inequalities involving integrals and the concept of average value of a function.

Definition: The average value of a function f on [a,b] is given by (1/(b-a)) ∫_a^b f(x)dx.

The page concludes with a list of important trigonometric identities:

Vocabulary:

cos(a+b) = cos(a)cos(b) - sin(a)sin(b)
cos(a-b) = cos(a)cos(b) + sin(a)sin(b)
sin(a+b) = sin(a)cos(b) + cos(a)sin(b)
sin(a-b) = sin(a)cos(b) - cos(a)sin(b)
cos(2a) = cos^2(a) - sin^2(a) = 1 - 2sin^2(a) = 2cos^2(a) - 1
sin(2a) = 2cos(a)sin(a)

These identities are crucial for solving complex trigonometric problems and are often used in calculus and physics applications.

This page serves as a valuable reference for students working with integrals and trigonometric functions, providing a concise summary of key properties and identities.

Inscris-toi pour voir le contenu. C'est gratuit!

Accès à tous les documents

Améliore tes notes

Rejoins des millions d'étudiants

En t'inscrivant, tu acceptes les Conditions d'utilisation et la Politique de confidentialité.

Derivatives and Primitives of Trigonometric Functions

This page introduces the fundamental concepts of derivatives and primitives of trigonometric functions, along with a key theorem on primitives.

The page begins with a table showing the derivatives and primitives of sine and cosine functions. This information is crucial for students learning calculus and trigonometry.

Highlight: The derivatives and primitives of sin(x) and cos(x) are presented in a clear, tabular format for easy reference.

A significant theorem is then presented and demonstrated:

Definition: Two primitives of the same function differ by a constant.

The proof of this theorem is provided, showing that if F and G are two primitives of a function f on an interval I, then F(x) - G(x) = C for all x in I, where C is a constant.

The page concludes with an introduction to differential equations of the form y' = ay.

Example: For differential equations of the form y' = ay, the solutions are functions of the form y(x) = Ce^(ax), where C is determined by initial conditions.

This information sets the foundation for more complex topics in differential equations and integration that will be covered in subsequent pages.

Rien ne te convient ? Explore d'autres matières.

Knowunity est la meilleure application scolaire dans cinq pays européens.

Knowunity a été mis en avant par Apple et a toujours été en tête des classements de l'App Store dans la catégorie Éducation en Allemagne, en Italie, en Pologne, en Suisse et au Royaume-Uni. Rejoins Knowunity aujourd'hui et aide des millions d'étudiants à travers le monde.

Chargement dans le

Google Play

Chargement dans le

App Store

4.9+

Note moyenne de l'appli

20 M

Les élèsves utilisent Knowunity

#1

Dans les palmarès des applications scolaires de 17 pays

950 K+

Les élèves publient leurs fiches de cours

Tu n'es toujours pas convaincu ? Regarde ce que disent les autres élèves ...

Louis B., utilisateur iOS

J'aime tellement cette application [...] Je recommande Knowunity à tout le monde ! !! Je suis passé de 11 à 16 grâce à elle :D

Stefan S., utilisateur iOS

L'application est très simple à utiliser et bien faite. Jusqu'à présent, j'ai trouvé tout ce que je cherchais :D

Lola, utilisatrice iOS