Integration –
Overall Objectives
Integrate ex, 1/x, sin x and cos x
Integrate by substitution and by parts
Find integrals of the form
f ' ( x)
1 1
dx dx dx
fx a x
2 2
a x
2 2
Integration – Lesson 1 Objectives
• Integrate ex, sin x and cos x
Integration
Remember
Differentiation and Integration are inverses
Multiply by power and reduce power by 1
dy
yx n
(n) x n 1
dx
Add 1 to power and divide by new power
n 1
x
x dx
n
c
n 1
Integration
Area under a curve between x=a and x = b
is given by the definite integral
b
f ( x)dx
a
Areas below the x axis are negative
ex
Differentiation
if f(x) = kex f `(x) = ke x
g`(x) = aeax
if g(x) = eax
Chain rule: u = ax du/dx = a Chain Rule
y = eu dy/du = eu dy dy du
dy/dx = eu x a = a eu = aeax
dx du dx
Integration
1 ax
e dx e c
ax
a
Trig Functions
Differentiation
y= sin x, dy/ = cos x
dx
y = cos x, dy/ = -sin x
dx
y= sin ax
u = ax du/dx = a Chain Rule
y = sin u dy/du = cos u dy dy du
dy/dx = cos u x a = a cos ax
dx du dx
y = cos ax
u = ax du/dx = a
y = cos u dy/du = -sin u
dy/dx = -sin u x a = -a sin ax
Trig Functions
Differentiation
y= sin ax, dy/ = a cos ax
dx
y = cos ax, dy/dx = -a sin ax
Integration 1
cos axdx a sin ax c
1
sin axdx cos ax c
a
Integration
Summary
1 ax
e dx e c
ax
a
1
cos axdx sin ax c
a
1
sin axdx cos ax c
a
Integration
• Now work through Page 107
• Follow example 1 then work through
Exercise A on pages 108 and 109