Class notes
Apuntes Calculo II (Técnicas de Integración)
- Course
- Institution
Son mis apuntes de mi curso de cálculo II
[Show more]
Seller
Follow
sebastinlepe
Content preview
Pontificada Universidad CatólicaMAT1126
Cálculo II
Autor:
Sebastián Lepe V.
23 de julio de 2021
, Cálculo II Sebastián Lepe V.
1. Técnicas de integración
En esta sección estudiaremos distintas técnicas para encontrar primitiva y valores de integrales,
estas técnicas giran en torno a la integración por parte y el cambio de variable, incluyendo susti-
tuciones trigonométricas y el uso de fracciones parciales.
1.1. Integración por parte y Cambio de variable
Teorema 3.1.1 (Integración por partes). Sean f, g : [a, b] → R funciones acotadas con deriva-
das integrables, entonces
Z b b
Z b
0
f (x)g (x) = f (x)g(x) − f 0 (x)g(x)
a a a
Dem. basta demostrar que que f (x)g(x) es una primitiva de f 0 (x)g(x) − f (x)g 0 (x). En efecto
Z b b
(f (x)g 0 (x) + f 0 (x)g(x))dx = f (x)g(x)
a a
luego despejando obtenemos el resultado
Ejemplo 3.1.1. Determine
Z
x sin(x)dx
Sol. Sea f (x) = x y g 0 (x) = sin(x), luego
Z Z
x sin(x)dx = − cos(x)x + cos(x)dx = − cos(x)x + sin(x) + C
Ejemplo 3.1.2. Determine
Z
log(x)dx
Sol. Sean f (x) = ln(x) y g 0 (x) = 1 entonces
Z Z
log(x)dx = log(x)x − dx = log(x)x − x + C
Ejemplo 3.1.3. Determine
1
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller sebastinlepe. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $3.49. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
66579 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now