# 8.1 Integration by Parts - PowerPoint PPT Presentation

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## 8.1 Integration by Parts

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### 8.2 Trigonometric Integrals ... 8.4 Integration of Rational Functions by Partial Fractions 8.8 Improper Integrals – PowerPoint PPT presentation

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Title: 8.1 Integration by Parts

1
????
• 8.1 Integration by Parts
• ?????
• 8.2 Trigonometric Integrals
• ???????
• 8.3 Trigonometric Substitution
• ?????
• 8.4 Integration of Rational Functions by Partial
Fractions
• ?????
• 8.8 Improper Integrals
• ???

2
8.2
• Trigonometric Integrals
• ???????

3
????
• ??????????
• ???????????????
• ???????????????

4
???????????1,2,3
• 1. sinmx?cosnx??
• m, n ?????
• m, n ????
• 2. tanmx?secnx??
• m ???
• n ???
• 3. sin mx?cos nx??

5
???????????1
• 1. sinmx?cosnx??
• m, n ?????????????u
• u sin x du cos xdx
• u cos x du -sin xdx
• m, n ?????????????
• sin2x (1-cos 2x)/2
• cos2x (1cos 2x)/2

6
Example 2
sin2 x 1- cos2x
u cos x
du -sin xdx
7
Example 3
• sin2x (1-cosx 2x)/2
• cos2x (1cos 2x)/2

8
p
0
0
p
9
Q 1
sin2 x 1- cos2x
u cos x
du -sin xdx
10
???????????2
• 2. tanmx?secnx??
• m ???
• u tan x du sec2 x dx
• tan2 x sec2 x - 1
• n ???
• u sec x du sec x tan x dx
• dv sec2 x dx v tan x

11
Example 5
u tan x
du sec 2x dx
12
Example 7
• u tan x du sec2 x dx
• tan2x sec2 x - 1

13
• u tan x du sec2 x dx

14
Q 2
tan2 x sec2 x - 1
15
Q 3
16
• u tan x du sec2 x dx

sec2x 1- tan2x
Q 3
17
Example 8
• u sec x du sec x tan x dx
• dv sec2 x dx v tan x

18
(No Transcript)
19
Example 8
20
???????????3
• 3. sin mx?cos nx??

21
Example 9
22
(No Transcript)
23
Q 4
24
?????????????
• 1. sinmx?cosnx??
• m, n ?????
• m, n ????
• 2. tanmx?secnx??
• m ???
• n ???
• 3. sin mx?cos nx??