Mathsoc Maths Refresher

7/31/2019 Mathsoc Maths Refresher

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Mathematics Refresher

PART 1

Indices Scientific Notation Engineering Notation Logarithms Radians Exercises

PART 2

Differentiation Integration

PART 1

Indices

The power of a number is indicated by an index, sometimes called an exponent. e.g.

a4

= a x a x a x a

Index Rules

Rule Example

am

.an

= a(m + n)

a3.a

5= a

8

n

m

a

a

= a(m - n)

4

6

a

a

= a2

(am

)n

= am.n

(a4)

2= a

8

a-m =m

a

1 a-6 =

1

a6

a0 = 1 40 = 1

a1/2

= a 91/2

= 9 = 3

a1/3

= 3 a 81/3

= 3 8 = 2


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Scientific Notation

This is also known as standard form. There is one digit before the decimal point and

the remaining significant figures after it. The number is multiplied by a power of 10 to

give it its correct value. e.g.

70 = 7 x 10

700 = 7 x 100 = 7 x 10 x 10 = 7 x 102

7200 = 7.2 x 1000 = 7.2 x 10 x 10 x 10 = 7.2 x 103

Small numbers are represented using negative powers of ten. e.g.

0.006 =

6

1000

=

6

10 x 10 x 10

=

6

103

= 6 x 10-3

0.000083 =

8.3

100000

=

8.3

10 x 10 x 10 x 10 x 10

=

8.3

105

= 8.3 x 10-5

0.000000092 =

9.2

100000000

=

9.2

10 x 10 x 10 x 10 x 10 x 10 x 10 x 10

=

9.2

108

= 9.2 x 10-8

This representation of numbers is known as scientific notation.

Examples

Represent the following numbers in scientific notation

0.000000102 = 1.02 x 10-7 1.045600000000 = 1.0456 x 100 189,000,000,000,000,000,000,000 = 1.89 x 1023 2,300,000 = 2.3 x 106

Engineering Notation

Engineering Notation is similar to scientific notation except the powers are restricted

to multiples of three and symbols are used to represent these powers. i.e.


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Y Yotta 1024

Z Zetta 1021

E Exa 1018

P Peta 1015

T Tera 1012

G Giga 109

M Mega 106

k kilo 103m milli 10

-3

micro 10-6

nano 10-9

p pico 10-12

f femto 10-15

a atto 10-18

z zepto 10-21

y yocto 10-24

v vimto 10-27

Examples

Number Scientific NotationEngineering Notation

45000000 4.5 x 107

45M

80000 8 x 104

80k

189000000000000000000000 1.89 x 1023

189Z

0.00000065 6.5 x 10-7

650

Logarithms

A logarithm is a any positive number which is expressed as a power of some arbitrary

number, which is called the base.

For example. We know that 24

= 16 Instead of using the number 16, we use the index

4, and call it a logarithm (or log) of 16. Provided that we know that we are dealing

with powers of 2 (the base number) we can always turn numbers into logs and logs

into ordinary numbers. Putting this in the form of an equation.

log216 = 4

Note the base number written as a subscript to the symbol log.

Logs to the base e (where e = 2.7182818) are known as natural logarithms. Normally

loge is written as ln e.g.

y = ex ln y = x


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Examples

Index Explanation Log

22 = 4 2 is the log of 4 to base 2 log24 = 2

e2

= 7.389 2 is the log of 7.389 to base e ln 7.389 = 2

102

= 100 2 is the log of 100 to base 10 log10100 = 2

bx

= y x is the log of y to base b logby = x

Log Rules

log (m x n) = log m + log n log (m/n) = log m - log n log mn = n log m log (1/m) = -log m log 1 = 0


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Examples

In each case solve for x

i. 3 = log6xx = 6

3

x = 216

ii. 1.7 = log2xx = 2

1.7

x = 3.249

iii. -2.8 = log9xx = 9

-2.8

x = 0.00213

iv. x = log91x = 0

v. x = logb1x = 0

vi. 3x = 7x.log103 = log107

x =

log107

log103

x =

0.8451

0.4771

x = 1.771

vii. 2 = logx9x

2= 9

x = 3


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viii. z = logb1 + logb(x/y) + logb(x.y) + logb(xy)z = 0 + logb(x) - logb(y) + logb(x) + logb(y) + y.logb(x)

z = 2.logb(x) + y.logb(x)

z = (2 + y)logb(x)

z

(2 + y)

= logb(x)

x = b(z / (2+y))

Using logs with a calculator

Most calculators only have logs to the base 10 and e. To overcome this problem the

following relationship is used.

logbx =

logyx

logyb

Example

Calculate log255

log255 =

log1055

log102

log255 =

1.74036

0.30103

log255 = 5.78136

Example

Calculate log430

log455 =

log1030

log104

log455 =

1.47712

0.60205

log455 = 2.4534


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Radians

An angle of one radian is formed when the length of the arc on the circumference of a

circle is equal to the radius of the circle. i.e.

= 1 rad

The constant is defined as the ratio of the diameter of a circle to its circumference.

i.e.

=c

d

= 3.141592654

where c is the circle's circumference and d is the circle's diameter.

The circumference of the circle can be defined as

c = 2r

Where r is the radius of the circle and c is the circumference

There are therefore 2. radius lengths in circumference of the circle. i.e

c

r

= 2.= 6.283185307

It therefore follows that there are 2. radians in a circle. i.e.


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Converting between radians and degrees

Consider the following two identical angles. The angle (d) in the left circle is

measured in degrees, while the angle (r) in the right circle is measured in radians.

It must hold that the ratio ofd to the complete circle is the same as the ratio ofr to

the complete circle. i.e

d

360

=

r

2

Example

Convert 32o

to radians.

r=32 x 2

360


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r= 0.5585 rad

Example

Convert 2.12 radians to degrees.

d=

2.12 x 360

2d= 121.5

o

Sinusoids

The term sinusoidal is used for waveforms defined by

f(t) = A.sin(.t + )

where A is the Amplitude, is the angular velocity, t is the independent time

variable, is the phase angle.

Exercises

1. IndicesSimplify

i. a5.a4ii. (a2.4)1.82

iii. 4m4.1 x 2.3m8.4iv. a-2.a4v.

15c4

3c5

vi. (3a3)2vii.

(3y2)

4

9y2


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viii. n3 x n-1 x n-2ix. (a4)x. 3 (27b6)

2. Scientific NotationRepresent the following numbers in scientific notation

i. 0.000000000006789ii. 0.1

iii. 4iv. 7800000000000000v. 30.2000000000

vi. 0.006734vii. 8000

viii. 0.00000356ix. 120100000000x. 2000.34

3. Engineering NotationRepresent the following numbers in Engineering Notation

i. 0.1ii. 8.67x103

iii. 0.707x102iv. 1.141x10-7v. 0.000000000006789

vi. 4vii.

1

6.78x105

ix.1

50x10-4

x. 7800000000000000xi. 30.2000000000

4. Logarithmsi. Evaluate.

a. log10 100b. log10 1000c. log10 1000000


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ii. For each of the expressions below, write the equivalent in logarithmicform.

a. 23 = 8b. 54 = 625c. 30 = 1d. 361/2 = 6e. 5-3 = 0.0008f. 25-1/2 = 0.2

iii. For each of the expressions below, write the equivalent in index form.a. log264 = 6

b. log31 = 0c. log55 = 1d. log464 = 3

iv. Simplify.a. log104 + log10 2

b. log1010 - log102c. 2.log103d. 0.5 log1016e.

log103 + 2.log107f. 3.log104 + 2log105 - log1020

v. Solve for xa. logx9 = 2

b. log28 = xc. 4x = 5d. 0.25x = 9

5. Radiansi. Convert the following angles to Radians

a. 50ob. 360oc. 120od. 6790oe. 180o

ii. Convert the following angles to degreesa. 5 Rad

b. /2 Radc. 3. Radd. 42 Rade. 73.89 Rad


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PART 2(a)

Differentiation

An Introduction ToAn Introduction ToAn Introduction ToAn Introduction ToDifferentiationDifferentiationDifferentiationDifferentiation

Reference:MathsDirectMathsDirectMathsDirectMathsDirect

Differentiation is the maths of changing variables. In science, what is reallyimportant, is not what value something has at the moment, but what value it will

have. This is where differentiation comes in. By differentiating a variable, we findthe rate at which it is changing. To begin with, we consider the gradients ofcurves, but soon move on, to apply differentiation to quantities changing withtime.

The Gradient Of A CurveThe Gradient Of A CurveThe Gradient Of A CurveThe Gradient Of A Curve

For a straight line,we defined thegradient as

This gradient wasconstant

If our graph is curved however, thenthe gradient will constantly bechanging.

The curves on the right are the same,but you can see that their gradients atthe two points are very different.

We define the gradient of a curve at a point, as being the gradient of the tangent to


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the curve at that point.

Clearly, this will change, depending on where you are on the curve. In otherwords, the gradient will be a function of x.

There is a special notation for

differentiation

If the curve has anequation connecting y &x

then the gradient iswritten

The most common type of expressionthat you will be asked to differentiate, isof the form

e.g.

There is avery simplerule, thatworks for anyvariation ofthis

=> where a & n are constants

i.e. You multiply by the power and thenreduce the power by one

Below are some very simple examples

=>

=>

=>

There are 2 special cases for this rule:

If the power of x is 1, then you are justleft with the number in front of it.

e.g. =>

=>


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If you have a number, on it's own, thisvanishes when you differentiate

e.g. =>

=>

The power of x does not have to bepositive integer

=>

=>

It is important thatbefore you applythe differentiationformula, you makesure that yourexpression iswritten in the

form:

You may need to re-arrange yourexpression, before you differentiate.

e.g.

Differentiate

First re-arrange

Then differentiate

Differentiate =>

Re-arrange


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Below are a few more examples of this basic formula

Differentiate =>

Differentiate =>

Differentiate =>


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If youdifferentiate

you get

Notice that the power of e does notchange after differentiation. Yousimply multiply by the number infront of the x

Examples

Differentiate =>

Differentiate =>

Differentiate =>

Stationary points:

There are threetypes of gradient a


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curve can have

In this case thecurve is increasing

Alternatively thecurve could bedecreasing

However, the mostinteresting case is.

At this point thecurve is stationarystationarystationarystationary.

This allows you tofind the maximumor minimum valueof a function, sincethese always occurwhere the gradientis 0

Stationary pointscan come in 3varieties


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A Minimum

A Maximum

A Point ofInflection

Below areexamples offinding the 3 typesof stationarypoints.

To find thestationary points,differentiate andsay that the resultmust equal 0.

Solve this equation,to find the value ofx at which thestationary pointoccurs.

Put this value of xinto yyyy.


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Therefore, thecurve has astationary point at

To find the natureof the stationarypoint, you look atthe gradient oneither side of the

point.

Gradient at x = -2

Gradient at x = 0

So the gradient isnegative to the leftof the point andpositive to theright.

This can berepresented by asketch

The point is aminimum.

Differentiate andsay that the resultis 0.

Solve the equationto find x.

Put this value intoy.

So the stationarypoint is at

To find the natureof the point, look atthe gradient oneither side of thepoint

The gradient ispositive to the leftand negative to theright

The point is amaximum


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Differentiate andsay that the resultis 0

Solve to find x andthe correspondingy.

Look at thegradient on eitherside of the point.

The gradient ispositive on bothsides of the point

This is a point ofinflection.

Note that a pointof inflection couldbe negative on bothsides of the point.

The Chain Rule

The function on the rightcould be considered acomposite function

where


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In differentiation we callthis "a function of afunction." The function ycould be written

where u is a function of x.

To differentiate such afunction, w.r.t. x, we usethe chain rulechain rulechain rulechain rule.

the du cancel out.

For example, to

differentiate the caseabove.

we say

where

We now differentiate thetwo expressionsseparately.

Now combine these, usingthe formula

For this example, we caneasily check that this iscorrect.

We can just expand thebrackets and differentiateas normal.

So the two methods agree.


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PRODUCT RULE

To differentiatea function

in the form

You should

use theformula

Example

Differentiate

First define u & v

Now differentiate u & vseparately

Now combine your results

Simplify

ResultResultResultResult =>

Quotient rule

To differentiatea function

in the form

You shoulduse the

formula

Example

Differentiate

First define u & v

Now differentiate u & vseparately


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Now combine yourresults

Simplify

ResultResultResultResult =>

IMPLICIT FUNCTION

So far we have differentiated functions in the form

so that y is expressed as a function of x. This is called explicit differentiation. Wenow want to look at differentiating expressions such as

where y is not given explicitly. This is called implicit differentiationimplicit differentiationimplicit differentiationimplicit differentiation....

To differentiate the above equation, w.r.t. x, you just differentiate each termseparately. This is straightforward for the x2 term and the 4, but how do youdifferentiate y2 w.r.t x?

In fact you use the chain rule. We can say that

Then applying the chain rule

we get

and we can work out that

so we have

We can now differentiate our original equation


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We could in this case, have re-arranged to make y the subject, then differentiated,but this would have been much more complicated, and in any case, there will beoccasions when it is not possible to make y the subject.

Let's consider a second example- Differentiate, w.r.t. x

Using the chain rule

so

The general rule for differentiating a y term w.r.t. x is, Diff the term w.r.t. y andDiff the term w.r.t. y andDiff the term w.r.t. y andDiff the term w.r.t. y andmultiply by dy/dxmultiply by dy/dxmultiply by dy/dxmultiply by dy/dx.


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Parametric differentiation

It is not always easy to write equations in the form

It is sometimes easier to write the relationship between x & y in terms of a thirdvariable, called a parameter. This parameter is usually represented by the letter t.

For example ;

So , when t = 3 ;

To find the gradient of a curve given parametrically, you have two options.

1 Rewrite the equation in cartesian form.

2 Differentiate parametrically, using the chain rule.

The problem with the first method is that equations are generally givenparametrically, specifically because they are hard to write in a cartesian form.

The example above, however is easy to convert, as an example.

Make t the subject of the xequation.

Substitute into the yequation

Differentiate as usual

In general, however, you should not use this method.

Parametric DifferentiationParametric DifferentiationParametric DifferentiationParametric Differentiation

To differentiate equations given parametrically, you should use the chain rule,

That is, you differentiate y and x separately, with respect to the parameter t. Thiscan be rewritten

In the example above


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This is the same answerthat we got before.

Another example

Find the gradient of the curve described by the parametric equations

This is an ellipse

In this case it would be difficult to write in a cartesian form. Differentiatingseparately gives.

Combining these gives

On the next page there is an example of finding the equation of a tangent to acurve given parametrically.

Trigonometric functions

This, and the derivatives of the other main trig functions are standard and may bequoted.

You will find them in the formula book, but you should try to learn them, as thiswill make solving problems muchmuchmuchmuch easier. The main derivatives are given below.


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The chain rule can be applied to all the trigonometric functions. Two examples aregiven below.

Define u

Differentiate u and y separately.

Combine the results and remove u.

Remember that this means all cubed.

Define u.

Differentiate u and y separately.

Combine the results and remove the u.

This is clearly a tedious procedure. The main results from using the chain rule ontrig functions are standard and can therefore be quoted. The main results are givenbelow.


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PART 2(a)

Integration

What Is Integration?What Is Integration?What Is Integration?What Is Integration? ReferenceMathsDirectMathsDirectMathsDirectMathsDirect

Suppose you were told that a function had a gradientgiven by

Could you work out that function?

By referring back to differentiation, you would notice that this gradient was givenby the function

Is this then our function?

Not necessarily! We would get the same gradient for the function


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This process of finding the function from it's gradient, is called IntegrationIntegrationIntegrationIntegration.

This is only the most basic use of integration. Together with differentiation, it isprobably the most important area of maths.

The basic method of integration, is to reverse differentiation.

To differentiate, wemultiplied by the power,then subtracted 1 from thepower

=>

To integrate, we reversethis exactly.

First add 1 to the power,then divide by the newnewnewnewpower.

=>

Notice that there is a constant added onto the function. This is because, as in thecase above, we do not know whether or not a number must be added to ourfunction.

ExampleExampleExampleExample

Find the curve whosegradient is

Add 1 the power an divideby new power.


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SimplifyImportantImportantImportantImportant If you leave offthe c, your answer is wrongand you willwillwillwill lose a mark

Clearly it is not very satisfactory to have these unknowns in our answers.

If we are told a point that the graph passes through, then we can evaluate theconstant.

For example

A curve has a gradient function

and passes through the point

Find the equation of the curve

Add 1 to the power and

divide by new power.Simplify

Substitutein

So the curve is

THE NOTATION OF INTEGRATION

Although integration hasbeen introduced as thereverse ofdifferentiation, it is animportant mathematicaloperation in it's ownright.

As such, it has it's ownspecial notation.

The symbol for

"integrate" isSo to integrate x, youwrite

This reads, "integrate xwith respect to x."

It is essential that youinclude the dx, as thistells you which variableyou want to integrate.The solution to theabove problem thenwould be


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If the expression thatyou are integratingcontains more than oneterm, then you mustbracket the terms, withthe dx outside thebrackets. e.g.

Another example

As with differentiation,it is important that allterms are in the correctform, before youintegrate. This mayinvolve some re-arranging.

DEFINITE INTEGRAL

Imagine a curve,whose gradient isgiven by

To find the differencein y, between x=1 andx=4, you would firstneed to integrate

When x=1

and when x=4

The change in y is,therefore

Notice that theunknown constant hasvanished.

When you find thedifference between anintegral at 2 values of


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x, you can ignore theconstant and are leftwith a number.

This is called DefiniteDefiniteDefiniteDefiniteIntegrationIntegrationIntegrationIntegration.

If you are going toevaluate a definiteintegral, you need tospecify limits. Thatis, you need to give 2values of x tointegrate between.

You give these limits,by writing the valuesat the top and bottomof the integral sign.The example above is

written:

The result of theintegration is writtenin square brackets,

with the limits to theright.

You then substitutethe limits into theintegral, the top limitfirst. Write the 2 partsin curly brackets, tokeep the separate.

Once you have workedthese out, subtract thesecond bracket fromthe first.

If you study thefollowing examples,the process shouldbecome clear.

Integrate 2x+3,between 0 and 5.

Evaluate thefollowing definiteintegral.

The terms are firstwritten in the form

ln1=0

ln 4 = ln 22 = 2 ln 2


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Evaluate thefollowing definiteintegral

Some Standard IntegralsSome Standard IntegralsSome Standard IntegralsSome Standard Integrals

Treating Integration as the reverse of differentiation,leads to some standard results. These results can all bequoted without any proof or derivation.

Below are some simple


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examples of thesestandard integrals.

Example 1Example 1Example 1Example 1






Integration Using IdentitiesIntegration Using IdentitiesIntegration Using IdentitiesIntegration Using Identities

Sometimes before integrating an expression, you needto re-arrange it. There are two main types of questionlike this.

Trigonometric expressions.

Partial Fractions.

You will often needto usetrigonometricidentities beforeintegrating. Forexample


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For this example,you first need theidentity.

This is now astandard integral

So the solution is.

A second example

First expand thebrackets.

Use an identity.The integral of sec2xcan be quoted.

The integral is

therefore.

To integrate somequotients, you needto split theexpressions intopartial fractions.

For example

which can bewritten as

Each part can beintegratedseparately.

Therefore theintegral is

A second example


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Split the expressioninto partial fractions.

Cross multiply.

Collect terms.

Equate thenumerators.

Form simultaneousequations and solveto find A and B.

Therefore thepartial fractionsare.

The integral cantherefore be written

So the answer is.

Integration By SubstitutionIntegration By SubstitutionIntegration By SubstitutionIntegration By Substitution

How do you integrate

Notice that one part isthe derivative of theother part.

If we make asubstitution,

then,

which re-arranges to

The integral thenbecomes.

which can beintegrated, w.r.t. u,

and then substituteback the xs.

This method ofsubstitution can be


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used for any integralwhere one part is thederivative of the other.

The two main formsare

and

For these cases yousubstitute as follows

Below are someexamples

Integrate

Use the substitution,

which gives.

The integral thenbecomes.

Integrate w.r.t. u

Substitute back thexs.

Integrate


which gives.

So the integralbecomes,

Integrate,

and substitute back thexs.

Integrate


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which gives

So the integralbecomes.

Integrate,

and replace the xs.


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Integration By PartsIntegration By PartsIntegration By PartsIntegration By Parts

How do you integratea product?

We saw that if one ofthe terms was thederivative of the other,i.e.

then we could use asubstitution

If this is not the case,however, you must usea method called

Integration by partsIntegration by partsIntegration by partsIntegration by parts.

This can be derivedfrom the product rulein differentiation.

Differentiate w.r.t. x.

Now integrate w.r.t. x

The left hand side justreturns to y, which wecan write as uv.

If we now re-arrange,we get.

This is the formula forintegration by parts.Note that one of theterms of the product istaken to be aderivative.

To see how this worksin practice, look at theexample below.

Integrate

First you must decidewhich term is thederivative and whichterm you are going todifferentiate. Ingeneral, differentiatethe x term.


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For the formula, youwill need tointegrate/differentiatethe terms separately.

Now just substitutethe above terms intothe formula.

You get anotherintegral, but this timeit is simple. Integrateit to get the finalanswer.

It may not always beso straight-forward

For example, integrate

Decide which terms tointegrate/differentiate

Substitute the termsinto the formula

We have been leftwith anotherproduct tointegrate. Thepower of x has,however, reducedby 1, and one moreintegration byparts will removeit.

You now need tointegrate

For the secondintegration, again

decide which terms tointegrate/differentiate.

You mustdifferentiate the sameterm as before,

otherwise you willjust return to whereyou started.

Put the above resultsinto the formula.

This time we get asimple integral.


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Now put this resultinto the originalintegral.

to get the finalanswer

One example ofintegration by partsthrows up a particulardifficulty.

Integrate.

In this case there is noobvious choice ofwhich term todifferentiate/integrate.

In fact, it does notmatter which way youdo it.

Using the abovechoices gives

We do not appear tohave gottenanywhere.

You now need topersevere, and performthe second integrationby parts.

You mustmustmustmustdifferentiate/integratethe same terms asbefore.

Again this does notappear to have helped.However, if wecombine the tworesults,

we get

Removing the

brackets, we see thatwe have the sameintegral on bothsides.

Re-arrange, to get

All h i i

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