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f ( g ( x ) ) g ( x ) d x = F ( g ( x ) ) + C .

Then

a b f [ g ( x ) ] g ( x ) d x = F ( g ( x ) ) | x = a x = b = F ( g ( b ) ) F ( g ( a ) ) = F ( u ) | u = g ( a ) u = g ( b ) = g ( a ) g ( b ) f ( u ) d u ,

and we have the desired result.

Using substitution to evaluate a definite integral

Use substitution to evaluate 0 1 x 2 ( 1 + 2 x 3 ) 5 d x .

Let u = 1 + 2 x 3 , so d u = 6 x 2 d x . Since the original function includes one factor of x 2 and d u = 6 x 2 d x , multiply both sides of the du equation by 1 / 6 . Then,

d u = 6 x 2 d x 1 6 d u = x 2 d x .

To adjust the limits of integration, note that when x = 0 , u = 1 + 2 ( 0 ) = 1 , and when x = 1 , u = 1 + 2 ( 1 ) = 3 . Then

0 1 x 2 ( 1 + 2 x 3 ) 5 d x = 1 6 1 3 u 5 d u .

Evaluating this expression, we get

1 6 1 3 u 5 d u = ( 1 6 ) ( u 6 6 ) | 1 3 = 1 36 [ ( 3 ) 6 ( 1 ) 6 ] = 182 9 .
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Use substitution to evaluate the definite integral −1 0 y ( 2 y 2 3 ) 5 d y .

91 3

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Using substitution with an exponential function

Use substitution to evaluate 0 1 x e 4 x 2 + 3 d x .

Let u = 4 x 3 + 3 . Then, d u = 8 x d x . To adjust the limits of integration, we note that when x = 0 , u = 3 , and when x = 1 , u = 7 . So our substitution gives

0 1 x e 4 x 2 + 3 d x = 1 8 3 7 e u d u = 1 8 e u | 3 7 = e 7 e 3 8 134.568.
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Use substitution to evaluate 0 1 x 2 cos ( π 2 x 3 ) d x .

2 3 π 0.2122

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Substitution may be only one of the techniques needed to evaluate a definite integral. All of the properties and rules of integration apply independently, and trigonometric functions may need to be rewritten using a trigonometric identity before we can apply substitution. Also, we have the option of replacing the original expression for u after we find the antiderivative, which means that we do not have to change the limits of integration. These two approaches are shown in [link] .

Using substitution to evaluate a trigonometric integral

Use substitution to evaluate 0 π / 2 cos 2 θ d θ .

Let us first use a trigonometric identity to rewrite the integral. The trig identity cos 2 θ = 1 + cos 2 θ 2 allows us to rewrite the integral as

0 π / 2 cos 2 θ d θ = 0 π / 2 1 + cos 2 θ 2 d θ .

Then,

0 π / 2 ( 1 + cos 2 θ 2 ) d θ = 0 π / 2 ( 1 2 + 1 2 cos 2 θ ) d θ = 1 2 0 π / 2 d θ + 0 π / 2 cos 2 θ d θ .

We can evaluate the first integral as it is, but we need to make a substitution to evaluate the second integral. Let u = 2 θ . Then, d u = 2 d θ , or 1 2 d u = d θ . Also, when θ = 0 , u = 0 , and when θ = π / 2 , u = π . Expressing the second integral in terms of u , we have

1 2 0 π / 2 d θ + 1 2 0 π / 2 cos 2 θ d θ = 1 2 0 π / 2 d θ + 1 2 ( 1 2 ) 0 π cos u d u = θ 2 | θ = 0 θ = π / 2 + 1 4 sin u | u = 0 u = θ = ( π 4 0 ) + ( 0 0 ) = π 4 .
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Key concepts

  • Substitution is a technique that simplifies the integration of functions that are the result of a chain-rule derivative. The term ‘substitution’ refers to changing variables or substituting the variable u and du for appropriate expressions in the integrand.
  • When using substitution for a definite integral, we also have to change the limits of integration.

Key equations

  • Substitution with Indefinite Integrals
    f [ g ( x ) ] g ( x ) d x = f ( u ) d u = F ( u ) + C = F ( g ( x ) ) + C
  • Substitution with Definite Integrals
    a b f ( g ( x ) ) g ' ( x ) d x = g ( a ) g ( b ) f ( u ) d u

Why is u -substitution referred to as change of variable ?

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2. If f = g h , when reversing the chain rule, d d x ( g h ) ( x ) = g ( h ( x ) ) h ( x ) , should you take u = g ( x ) or u = h ( x ) ?

u = h ( x )

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In the following exercises, verify each identity using differentiation. Then, using the indicated u -substitution, identify f such that the integral takes the form f ( u ) d u .

x x + 1 d x = 2 15 ( x + 1 ) 3 / 2 ( 3 x 2 ) + C ; u = x + 1

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x 2 x 1 d x ( x > 1 ) = 2 15 x 1 ( 3 x 2 + 4 x + 8 ) + C ; u = x 1

f ( u ) = ( u + 1 ) 2 u

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Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
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Akash Reply
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Maciej
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Abigail
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Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
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Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
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what's the easiest and fastest way to the synthesize AgNP?
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China
Cied
types of nano material
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I start with an easy one. carbon nanotubes woven into a long filament like a string
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many many of nanotubes
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Leaves accumulate on the forest floor at a rate of 2 g/cm2/yr and also decompose at a rate of 90% per year. Write a differential equation governing the number of grams of leaf litter per square centimeter of forest floor, assuming at time 0 there is no leaf litter on the ground. Does this amount approach a steady value? What is that value?
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Practice Key Terms 2

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Source:  OpenStax, Calculus volume 2. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11965/1.2
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