Integration Overall Objectives

1
Integration Overall Objectives

• Integrate ex, 1/x, sin x and cos x
• Integrate by substitution and by parts
• Find integrals of the form

2
Integration Lesson 1 Objectives

• Integrate ex, sin x and cos x

3
Integration

• Remember
• Differentiation and Integration are inverses
• Multiply by power and reduce power by 1
• Add 1 to power and divide by new power

4
Integration

• Area under a curve between xa and x b is given
  by the definite integral
• Areas below the x axis are negative

5
ex
Differentiation
f (x) kex
if f(x) kex
g(x) aeax
if g(x) eax
Chain rule u ax du/dx a
y eu dy/du eu
dy/dx eu x a a eu aeax
Integration
6
Trig Functions
Differentiation

• y sin x, dy/dx cos x
• y cos x, dy/dx -sin x

y sin ax y cos ax
u ax du/dx a
y sin u dy/du cos u
dy/dx cos u x a a cos ax
u ax du/dx a
y cos u dy/du -sin u
dy/dx -sin u x a -a sin ax
7
Trig Functions
Differentiation
y sin ax, dy/dx a cos ax y cos ax, dy/dx
-a sin ax
Integration
8
Integration

• Summary

9
Integration

• Now work through Page 107
• Follow example 1 then work through Exercise A on
  pages 108 and 109