21
Introduction
• Integration is the reverse process of Differentiation
• Differentiating gives us a formula for the gradient
• Integrating can get us the formula for the curve, if we know the gradient function
• It can also be used to calculate the Area under a curve
IntegrationYou can integrate functions of the form f(x) = axn where ‘n’ is
real and ‘a’ is a constant
Integrating is the reverse process of differentiation. Let us think about a differentiation for a moment.
If:
If:
If:
8A
2y x
2dy
xdx
2 5y x
2dy
xdx
2 - 7y x
2dy
xdx
So integrating 2x should give us x2, but we will be unsure as to whether a number has been added or
taken away
Differentiating
Function
Multiply by the power
Reduce the power by 1
Gradient Function
Integrating
Function
Divide by the power
Increase the power by 1
Gradient Function
IntegrationYou can integrate functions of the form f(x) = axn where ‘n’ is
real and ‘a’ is a constant
Integrating is the reverse process of differentiation. Let us think about a differentiation for a moment.
8A
Integrating
Function
Divide by the power
Increase the power by 1
Gradient Function
Mathematically speaking…
ndyx
dx
1
1
nxy
n
We increased the power by 1, then divided by the
(new) power
If:
Then:
IntegrationYou can integrate functions of the form f(x) = axn where ‘n’ is
real and ‘a’ is a constant
Integrating is the reverse process of differentiation. Let us think about a differentiation for a moment.
8A
Integrating
Function
Divide by the power
Increase the power by 1
Gradient Function
Example Questions
ndyx
dx
1
1
nxy
n
If:
Then:
Integrate the following:
a)4dyx
dx
5
5
xy c
Increase the power by one, and divide by the new power
DO NOT FORGET TO ADD C!
b)5dy
xdx
4
4
xy
c
Increase the power by one, and divide by the new power
DO NOT FORGET TO ADD C!
41
4y x c
IntegrationYou can integrate functions of the form f(x) = axn where ‘n’ is
real and ‘a’ is a constant
Integrating is the reverse process of differentiation. Let us think about a differentiation for a moment.
8A
Integrating
Function
Divide by the power
Increase the power by 1
Gradient Function
Example Questions
ndyx
dx
1
1
nxy
n
If:
Then:
Integrate the following:
c)32
dyx
dx
42
4
xy c
Increase the power by one, and divide by the new power
DO NOT FORGET TO ADD C!
d)1
23dy
xdx
3
233
2
xy c
Increase the power by one, and divide by the new power
DO NOT FORGET TO ADD C!
4
2
xy c
3
22y x c
IntegrationYou can apply the idea of
Integration separately to each term of dy/dx
In short, if you have multiple terms to integrate, do them all separately
8B
Example QuestionIntegrate the following:
13 26 2 3
dyx x x
dx
26
2
xy
22
2
x
3
23
32
x
23y x 2 x3
2 2x c
Integrate each part separately
‘Tidy up’ terms if possible
IntegrationYou can apply the idea of
Integration separately to each term of dy/dx
In short, if you have multiple terms to integrate, do them all separately
8B
Example QuestionIntegrate the following:
1 3
2 21 1
'( )2 2
f x x x
1
2121
2
xy
1
212
12
x
Integrate each part separately
1
2y x1
2 x
y x1
x
Deal with the
fractions
Rewrite if necessary
c
c
IntegrationYou need to be able to use
the correct notation for Integration
8C
nx dx
This the the integral sign,
meaning integrate
This is the expression to be
integrated (brackets are often used to separate
it)
The dx is telling you to integrate ‘with respect to
x’
Example QuestionFind:
132( 2 ) x x dx
3
2
3
2
x
42
4
x
3
22
3x 41
2x c
Integrate each part separately
Deal with the
fractions
IntegrationYou need to be able to use
the correct notation for Integration
8C
nx dx
This the the integral sign,
meaning integrate
This is the expression to be
integrated (brackets are often used to separate
it)
The dx is telling you to integrate the
‘x’ parts
Example QuestionFind:
3
2( 2) x dx
1
2
1
2
x
2x
1
2 2x
2x c
Integrate each part separately
Deal with the
fractions
IntegrationYou need to be able to use
the correct notation for Integration
8C
nx dx
This the the integral sign,
meaning integrate
This is the expression to be
integrated (brackets are often used to separate
it)
The dx is telling you to integrate the
‘x’ parts
Example QuestionFind:
2 2 2(3 ) x px q dx
33
3
x
1
1
px
Integrate each part separately
2 q x
3 x 1 px 2 q x c
Deal with the
fractions
p and q2 should be treated as if they were just numbers!
IntegrationYou need to be able to use
the correct notation for Integration
8C
nx dx
This the the integral sign,
meaning integrate
This is the expression to be
integrated (brackets are often used to separate
it)
The dx is telling you to integrate the
‘x’ parts
Example QuestionFind:
2(4 6) t dt
34
3
t 6t
Integrate each part separately
c
IntegrationYou can find the constant of
integration, c, if you are given a point that the function passes
through
Up until now we have written ‘c’ when Integrating.
The point of this was that if we differentiate a number on its own, it disappears.
Consequently, when integrating, we cannot be sure whether a number was there originally, and what it was if there was one…
Step 1: Integrate as before, putting in ‘c’Step 2: Substitute the coordinate in, and work out what ‘c’ must be to make the equation balance.. 8E
Example QuestionThe curve X with equation y = f(x)
passes through the point (2,15). Given that:
Find the equation of X.
2'( ) 5 3f x x x
2'( ) 5 3f x x x Integrate
3 25 3( )
3 2
x xf x c
Sub in (2,15)3 25(2) 3(2)
153 2
c
40 1215
3 2c
115 19
3c
1 4
3c
3 25 3 1( ) 4
3 2 3
x xf x
Work out each fraction
Add the fractions together
Work out c
IntegrationYou can find the constant of
integration, c, if you are given a point that the function passes
through
Up until now we have written ‘c’ when Integrating.
The point of this was that if we differentiate a number on its own, it disappears.
Consequently, when integrating, we cannot be sure whether a number was there originally, and what it was if there was one…
Step 1: Integrate as before, putting in ‘c’Step 2: Substitute the coordinate in, and work out what ‘c’ must be to make the equation balance.. 8E
Example QuestionThe curve X with equation y = f(x)
passes through the point (4,5). Given that:
Find the equation of X.
2 2'( )
xf x
x
2 2'( )
xf x
x
2
'( )x
f xx
2
x
3
2'( )f x x1
2 2x
5
2
( )5
2
xf x
1
22
12
x
5
22
( )5
f x x1
2 4x + c
Split into 2 parts
Write in the form axn
Integrate
IntegrationYou can find the constant of
integration, c, if you are given a point that the function passes
through
Up until now we have written ‘c’ when Integrating.
The point of this was that if we differentiate a number on its own, it disappears.
Consequently, when integrating, we cannot be sure whether a number was there originally, and what it was if there was one…
Step 1: Integrate as before, putting in ‘c’Step 2: Substitute the coordinate in, and work out what ‘c’ must be to make the equation balance.. 8E
Example QuestionThe curve X with equation y = f(x)
passes through the point (4,5). Given that:
Find the equation of X.
2 2'( )
xf x
x
5
22
( )5
f x x1
2 4x + c
52( )
5f x x 4 x + c
525 4
5 4 4 + c
5 12.8 8 + c
0.2 c
Rewrite for substitution
y = 5, x = 4
Work out each part carefully
5
22
( )5
f x x1
2 4x + 0.2
Summary
• We have learnt what Integration is
• We have seen it combined with rewriting for substitution
• We have learnt how to calculate the missing value ‘c’, and why it exists in the first place
