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“Teach A Level Maths” Vol. 1: AS Core Modules

24: Indefinite Integration

© Christine Crisp

Indefinite Integration

Module C1

Module C2

AQA

MEI/OCR

Edexcel

OCR

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Indefinite Integration We first need to consider an example of differentiation e.g.1 Differentiate (a) y  x 2  3

(a) y  x 2  3 

dy

(b) y  x  1 2

 2x

dx

Equal !

(b) y  x 2  1  dy  2 x dx

The gradient functions are the same since the graph of y  x 2  1 is a just a translation of

y x 3 2

Indefinite Integration Graphs of the functions y x 3 2

e.g. the gradient at x = 1 is 2

y  x 1 2

At each value of x, the gradients of the 2 graphs are the same dy

dx

 2x

Indefinite Integration Indefinite integration is the reverse of differentiation If we are given the gradient function and want to find the equation of the curve, we reverse the process of differentiation BUT the constant is unknown So,

dy dx

 2x

 y  x C 2

C is called the arbitrary constant or constant of integration The equation y  x 2  C forms a family of curves

Indefinite Integration e.g.2 Find the equation of the family of curves which have a gradient function given by Solution:

dy

dy

 6x

2

dx  6x

2

dx

To reverse the rule of differentiation: 

y  6x

3

• •

add 1 to the power divide by the new power

Indefinite Integration e.g.2 Find the equation of the family of curves which have a gradient function given by Solution:

dy

dy

 6x

2

dx  6x

2

dx

To reverse the rule of differentiation: 2



y

6x

3

C

31



y  2x  C 3

• •

add 1 to the power divide by the new power



add C

Tip: Check the answer by differentiating

Indefinite Integration The graphs look like this:

The gradient function dy

 6x

dy

dy

 6x

2

dx

dx

2

dx y



y  2x  C 3

( Sample of 6 values of C )

y  2x  5 3

y  2x

3

Indefinite Integration e.g. 3 Find the equation of the family of curves with gradient function dy

 3x  1

dx

Solution: The index of x in the term 3x is 1, so adding 1 to the index gives 2. The constant 1 has no x. It integrates to x. 

y

3x

2

x C

2

We can only find the value of C if we have some additional information

Indefinite Integration Exercises Find the equations of the family of curves with the following gradient functions: 1.

dy

 3x  4x 2

dx

1

Ans :

2. 3.

dy

y

3x

2

3



1 3

 x  2

1

x 1

dx

2

dy

 ( x  2)( x  3)

dx

4x

2

C 

y  x  2x  C 3

2

21 Ans :

y

x

3

3



x

2

 xC

4

N.B. Multiply out the brackets first

Indefinite Integration Exercises Find the equations of the family of curves with the following gradient functions: 1.

dy

 3x  4x 2

dx

1

Ans :

2.

dy

y

dy

2

3



4x

1 3

 x  2

dx

3.

3x

1

2

C 

Ans :

x 1

 ( x  2)( x  3)

dx

y

3

2

21 y

x



3

3



x

2

2

x

3

3

2

Ans :

y  x  2x  C

dy



x

2

 xC

4

 x  x6

dx

 6x  C

2

Indefinite Integration Finding the value of C

e.g.1 Find the equation of the curve which passes through the point (1, 2) and has gradient function given by dy 2  x x2

dx

Solution:

dy dx

 x x2  2

y

x

3

3



x

2

2

 2x  C

Indefinite Integration Finding the value of C

e.g.1 Find the equation of the curve which passes through the point (1, 2) and has gradient function given by dy 2  x x2

dx

Solution:

dy dx

 x x2  2

y



6 is the common denominator

2



3

2

1 3

C 



1

1 6



1

2C

So,

1

2

 2x  C

 2 (1)  C

2 23



2

x

2

3

(1, 2) is on the curve:



x

3

C

6

y

x

3

3



x

2

2

 2x 

1 6

Indefinite Integration Exercises 1. Find the equation of the curve with gradient function dy dx



1

x which passes through the point ( 2, -2 )

2

2. Find the equation of the curve with gradient function dy dx

 ( x  1)( x  2) which passes through the point ( 2, 1 )

Indefinite Integration Solutions

1.

dy dx



1

x

2

Ans:

x

y

2

C

4

( 2, -2 ) lies on the curve 

2

( 2)

2

C

4



21  C  So,

1  C y

x

2

4

1

Indefinite Integration Solutions

2.

dy

 ( x  1)( x  2)



dx

 ( 2, 1 ) on the curve

y

3

x



3x

3



1

 1

( 2)

3

dx

2

 2x  C



3( 2)





So,

2

 2( 2)  C

35 3

3 y

3

3



3x 2

2

2

64  C x

 x  3x  2

2

3

8

dy

2

 2x 

35 3

C

Indefinite Integration Notation for Integration dy

e.g. 1 We know that

dx

1

 2x

 y

Another way of writing integration is:



2x

2

C

21

2 x dx  x  C

Called the integral sign

2

We read this as “d x ”. It must be included to indicate that the variable is x

In full, we say we are integrating “ with respect to x “.

Indefinite Integration e.g. 2 Find (a)



(b)

3 dx

 3 dt

Solution: (a) ( Integrate with respect to x )

 3 dx  3 x  C  3 dt  3t  C

(b) ( Integrate with respect to t )

e.g. 3 Integrate x 3  x 2  2 x  1 with respect to x Solution: The notation for integration must be written We have done the integration so there is no integral sign

  

x  x  2 x  1 dx 3

x

2

4

4 4 x 4

 

x

3

3 3 x 3

1



2x

2

 xC

21  x  xC 2

Indefinite Integration Exercises 1. Find

(a)



3 x  4 x  2 dx

(b)



4t  8t  4t  3 dt

Ans : x  2 x  2 x  C 3

2

3

2

2

Ans : t  4

8t

3

 2t  3t  C 2

3

2. Integrate the following with respect to x: (b) 4 x 3  9 x 2  6 x  7

(a) x 2  3 x  2 Ans : (a)



(b)



x  3 x  2 dx  2

x

3

3



3x

2

 2x  C

2

4 x  9 x  6 x  7 dx  x  3 x  3 x  7 x  C 3

2

4

3

2

Indefinite Integration Summary

 Indefinite Integration is the reverse of differentiation.  A constant of integration, C, is always included.  Indefinite Integration is used to find a family of curves.  To find the curve through a given point, the value of C is found by substituting for x and y.  There are 2 notations: e.g.

e.g.

dy

 4x

3



y  x C 4

dx



4x

3

dx  x  C 4

Indefinite Integration

Indefinite Integration

The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.

For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

Indefinite Integration Indefinite integration is the reverse of differentiation

If we are given the gradient function and want to find the equation of the curve, we reverse the process of differentiation BUT the constant is unknown So,

dy dx

 2x

 y  x C 2

C is called the arbitrary constant or constant of integration The equation y  x 2  C forms a family of curves

Indefinite Integration e.g.

Find the equation of the family of curves which have a gradient function given by

Solution:

dy

dy

 6x

2

dx  6x

2

dx

To reverse the rule of differentiation: 2



y

6x

3

C

31



y  2x  C 3

• •

add 1 to the power divide by the new power



add C

Tip: Check the answer by differentiating The graphs look like this:

Indefinite Integration The gradient function dy

 6x

dy

 6x

dy

2

dx

dx

2

dx y



y  2x  C 3

( Sample of 6 values of C )

y  2x  5 3

y  2x

3

We can only find the value of C if we have some additional information

Indefinite Integration Finding the value of C e.g.1 Find the equation of the curve which passes through the point (1, 2) and has gradient function given by dy 2  x x2

dx

Solution:

dy dx

 x x2  2

(1, 2) is on the curve: 6 is the common denominator 

2

1 3

C 

1 6





1

y

3



x

2

 2x  C

2 3 1 1 2    2 (1)  C 3 2

2C

23



2

So,

x

C

6

y

x

3

3



x

2

2

 2x 

1 6

Indefinite Integration e.g. 2 Find the equation of the family of curves with gradient function dy

 3x  1

dx

Solution: The index of x in the term 3x is 1, so adding 1 to the index gives 2. The constant 1 has no x. It integrates to x. 

y

3x 2

2

x C

Indefinite Integration Notation for Integration dy

e.g. 1 We know that

dx

1

 2x

 y

Another way of writing integration is:



2x

2

C

21

2 x dx  x  C

Called the integral sign

2

We read this as “d x ”. It must be included to indicate that the variable is x

In full, we say we are integrating “ with respect to x “.

Indefinite Integration Summary  Indefinite Integration is the reverse of differentiation.  A constant of integration, C, is always included.

 Indefinite Integration is used to find a family of curves.  To find the curve through a given point, the value of C is found by substituting for x and y.  There are 2 notations: e.g.

e.g.

dy

 4x

3



y  x C 4

dx



4x

3

dx  x  C 4

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