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Use substitution to find the antiderivative of 3 x 2 ( x 3 3 ) 2 d x .

3 x 2 ( x 3 3 ) 2 d x = 1 3 ( x 3 3 ) 3 + C

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Sometimes we need to adjust the constants in our integral if they don’t match up exactly with the expressions we are substituting.

Using substitution with alteration

Use substitution to find the antiderivative of z z 2 5 d z .

Rewrite the integral as z ( z 2 5 ) 1 / 2 d z . Let u = z 2 5 and d u = 2 z d z . Now we have a problem because d u = 2 z d z and the original expression has only z d z . We have to alter our expression for du or the integral in u will be twice as large as it should be. If we multiply both sides of the du equation by 1 2 . we can solve this problem. Thus,

u = z 2 5 d u = 2 z d z 1 2 d u = 1 2 ( 2 z ) d z = z d z .

Write the integral in terms of u , but pull the 1 2 outside the integration symbol:

z ( z 2 5 ) 1 / 2 d z = 1 2 u 1 / 2 d u .

Integrate the expression in u :

1 2 u 1 / 2 d u = ( 1 2 ) u 3 / 2 3 2 + C = ( 1 2 ) ( 2 3 ) u 3 / 2 + C = 1 3 u 3 / 2 + C = 1 3 ( z 2 5 ) 3 / 2 + C .
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Use substitution to find the antiderivative of x 2 ( x 3 + 5 ) 9 d x .

( x 3 + 5 ) 10 30 + C

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Using substitution with integrals of trigonometric functions

Use substitution to evaluate the integral sin t cos 3 t d t .

We know the derivative of cos t is sin t , so we set u = cos t . Then d u = sin t d t . Substituting into the integral, we have

sin t cos 3 t d t = d u u 3 .

Evaluating the integral, we get

d u u 3 = u −3 d u = ( 1 2 ) u −2 + C .

Putting the answer back in terms of t , we get

sin t cos 3 t d t = 1 2 u 2 + C = 1 2 cos 2 t + C .
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Use substitution to evaluate the integral cos t sin 2 t d t .

1 sin t + C

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Sometimes we need to manipulate an integral in ways that are more complicated than just multiplying or dividing by a constant. We need to eliminate all the expressions within the integrand that are in terms of the original variable. When we are done, u should be the only variable in the integrand. In some cases, this means solving for the original variable in terms of u . This technique should become clear in the next example.

Finding an antiderivative using u -substitution

Use substitution to find the antiderivative of x x 1 d x .

If we let u = x 1 , then d u = d x . But this does not account for the x in the numerator of the integrand. We need to express x in terms of u . If u = x 1 , then x = u + 1 . Now we can rewrite the integral in terms of u :

x x 1 d x = u + 1 u d u = u + 1 u d u = ( u 1 / 2 + u −1 / 2 ) d u .

Then we integrate in the usual way, replace u with the original expression, and factor and simplify the result. Thus,

( u 1 / 2 + u −1 / 2 ) d u = 2 3 u 3 / 2 + 2 u 1 / 2 + C = 2 3 ( x 1 ) 3 / 2 + 2 ( x 1 ) 1 / 2 + C = ( x 1 ) 1 / 2 [ 2 3 ( x 1 ) + 2 ] + C = ( x 1 ) 1 / 2 ( 2 3 x 2 3 + 6 3 ) = ( x 1 ) 1 / 2 ( 2 3 x + 4 3 ) = 2 3 ( x 1 ) 1 / 2 ( x + 2 ) + C .
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Use substitution to evaluate the indefinite integral cos 3 t sin t d t .

cos 4 t 4 + C

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Substitution for definite integrals

Substitution can be used with definite integrals, too. However, using substitution to evaluate a definite integral requires a change to the limits of integration. If we change variables in the integrand, the limits of integration change as well.

Substitution with definite integrals

Let u = g ( x ) and let g be continuous over an interval [ a , b ] , and let f be continuous over the range of u = g ( x ) . Then,

a b f ( g ( x ) ) g ( x ) d x = g ( a ) g ( b ) f ( u ) d u .

Although we will not formally prove this theorem, we justify it with some calculations here. From the substitution rule for indefinite integrals, if F ( x ) is an antiderivative of f ( x ) , we have

Questions & Answers

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I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
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what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
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sciencedirect big data base
Ernesto
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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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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