Docsity
Docsity
Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity

Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan

Guidelines and tips
Guidelines and tips
Sell on Docsity
Sell on Docsity
Docsity AI
Docsity AI
Log inSign up

Prepare for your exams

Study with the several resources on Docsity

Find documents

Prepare for your exams with the study notes shared by other students like you on Docsity

Search for your university

Find the specific documents for your university's exams

Docsity AINEW

Summarize your documents, ask them questions, convert them into quizzes and concept maps

Explore questions

Clear up your doubts by reading the answers to questions asked by your fellow students

Earn points to download

Earn points by helping other students or get them with a premium plan

Share documents
download point icon20 Points

For each uploaded document

Answer questions
download point icon5 Points

For each given answer (max 1 per day)

All the ways to get free points
Get points immediately

Choose a premium plan with all the points you need

Study Opportunities

Choose your next study program

Get in touch with the best universities in the world. Search through thousands of universities and official partners

Community

Ask the community

Ask the community for help and clear up your study doubts

Free resources

Our save-the-student-ebooks!

Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors

Techniques of Integration: Integration by Substitution and Integration by Parts, Lecture notes of Statistics

University of BotswanaStatistics

This chapter covers two techniques for finding indefinite integrals: integration by substitution and integration by parts. The former is used when the integral can be transformed into the derivative of a known function, while the latter is used when the integral can be expressed as the product of two functions. Examples and solutions for each technique.

Typology: Lecture notes

2021/2022

Uploaded on 01/28/2022

tmkgomotso
tmkgomotso 🇧🇼

4.8

(19)

68 documents

1 / 2

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
Chapter 6 - Techniques of Integration
6.1 Integration by Substitution
We know that the derivative of a function is obtained by the Chain rule:
, that is let , then
. Therefore, if we have a function of the form , then the
indefinite integral (anti-derivative) is , that is
.
Example 6.1.1
Find .
Solution 6.1.1
Put . Therefore, this implies that
.
Example 6.1.2
(i) Evaluate (i) and (ii) .
Solution 6.1.2
(i) Put . Therefore, this implies that
= .
(ii)
Example 6.1.3
pf2

Partial preview of the text

Download Techniques of Integration: Integration by Substitution and Integration by Parts and more Lecture notes Statistics in PDF only on Docsity!

Chapter 6 - Techniques of Integration

6.1 Integration by Substitution

We know that the derivative of a function is obtained by the Chain rule: , that is let , then

. Therefore, if we have a function of the form , then the indefinite integral (anti-derivative) is , that is . Example 6.1. Find. Solution 6.1. Put. Therefore, this implies that . Example 6.1. (i) Evaluate (i) and (ii). Solution 6.1. (i) Put. Therefore, this implies that =. (ii) Example 6.1.

, put. This implies that . Example 6.1. Evaluate , put. We know that when

. This implies that .

6.2 Integration by parts

Suppose and are functions of , that is and , then . Example 6.2. Evaluate i) ii) Solution 6.2.