8.1 Integration by Parts - PowerPoint PPT Presentation

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Title:

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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Date added: 15 January 2020
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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??
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