[email protected] MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 1 Bruce Mayer, PE Chabot...

[email protected] • MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

§5.2 Integration

By Substitution



Review §

Any QUESTIONS About• §5.1 → AntiDerivatives

Any QUESTIONS About HomeWork• §5.1 → HW-22

5.1

http://www.google.com/url?sa=i&rct=j&q=antiderviative&source=images&cd=&cad=rja&docid=c3KsnpWfTtQFpM&tbnid=5CmMWOtI4stg8M:&ved=0CAUQjRw&url=http://ibscrewed4maths.wordpress.com/&ei=EcntUd3pJcHXygGZ3oCoAQ&bvm=bv.49641647,d.aWc&psig=AFQjCNEKikHlFY5i363oZZM9y6C8unJvTQ&ust=1374624228835574

http://www.google.com/url?sa=i&rct=j&q=antiderivative%20symbol&source=images&cd=&cad=rja&docid=e4mT9pBhGxpiBM&tbnid=hWllvSFluip6mM:&ved=0CAUQjRw&url=http://www.math.fsu.edu/~wooland/calculus/L16/Lecture16.html&ei=Xc3tUfT9GIuErQGvqoGgBg&bvm=bv.49478099,bs.1,d.aWc&psig=AFQjCNGbCEGiBfkYlWq2SCPqQAiLlhep1g&ust=1374625479303976



§5.2 Learning Goals

Use the method of substitution to find indefinite integrals

Solve initial-value and boundary-value problems using substitution

Explore a price-adjustment model in economics

http://www.google.com/url?sa=i&rct=j&q=antidervative%20substitution&source=images&cd=&cad=rja&docid=wMlaADPodezdzM&tbnid=z6SXKGb28yDMWM:&ved=0CAUQjRw&url=http://www.tumblr.com/tagged/antiderivative&ei=zMvtUfDaH-LXygGDooDoBw&psig=AFQjCNE2GHGxqQtPEeN5QK0gJkfe1aEs9g&ust=1374625030913651



Recall: Fcn Integration Rules

1. Constant Rule: for any constant, k

2. Power Rule:for any n≠−1

3. Logarithmic Rule:for any x≠0

4. Exponential Rule:for any constant, k

Cxkdxk

Cn

xdxx

nn

1

1

Cxdxx

ln 1

Cek

dxe kxkx 1



Recall: Integration Algebra Rules

1. Constant Multiple Rule: For any constant, a

2. The Sum or Difference Rule:

• This often called the Term-by-Term Rule

dxxuudxxua

dxxvdxxudxxvxu

dxxqdxxpdxxqxp



Integration by Substitution

Sometimes it is MUCH EASIER to find an AntiDerivative by allowing a new variable, say u, to stand for an entire expression in the original variable, x

In the AntiDerivative expression ∫f(x)dx substitutions must be made:• Within the Integrand• For dx

Along Lines →

dxxgdu

xgdx

du

xGdx

du

dx

dxGu

Let



Investigate Substitution

Compute the family of AntiDerivatives given by

a. by expanding (multiplying out) and using rules of integration from Section 5.1

b. by writing the integrand in the form u2and guessing at an antiderivative.

dxx 13 2




SOLUTION a: “Expand the BiNomial” by “FOIL”

Multiplication

SOLUTION b: Let: Sub u into Expression →

dxxxxdxxxdxx 1339 1313 13 22

C 33 169 23Rules Sum &Power

2 xxxdxxx

13 xu

dxu 2




Examine the “substituted” expression to find the• Integrand stated in terms of u• Integrating factor (dx) stated in terms of x

The Integrand↔IntegratingFactor MisMatch does Not Permit the AntiDerivation to move forward.• Let’s persevere, with the understanding is

something missing by flagging that with a (well-placed) question mark.

dxudxu 22




Continuing

)31?()399(? 123 Cxxx

123 31?&)399(? CCCxxx




The integral in part (b) (which is speculative) agrees with the integral calculated in part (a) (using established techniques) when

By Correspondence observe that ?=⅓• This Begs the Question: is there some

systematic, a-priori, method to determine the value of the question-mark?



SubOut Integrating Factor, dx

Let the single value, u, represent an algebraic expression in x, say:

Then take thederivative of bothsides

Then Isolate dx

xGu

orxGudx

d

xGdx

du

dx

d

xgdx

du xg

dxxg

dx

du




Then the Isolated dx:

Thus the SubStitution Components

Consider the previous example

Let: Then after subbing:

dxxg

du

dxxdG

du

xg

dudxxGu and

dxx 13 2

13 xxGu

dxu 2




Now Use Derivation to Find dx in terms of du →

Multiply both sides by dx/3 to isolate dx

Now SubOut Integrating Factor, dx

Now can easily AntiDerivate (Integrate)

13 xdx

du

dx

d

dxdu

dx

dudx

dx

du

33

303

duudu

uduudxu 2

222

3

1

33




Integrating

Recall: BackSub u=3x+1 into integration result

Expanding the BiNomial find

3

&93933

1

3

1 32 K

CCuKu

Ku

duu

13 xu

C

xC

uduudxx

9

13

33

3

1 13

3322

CxxxC

xxxC

x

9

133

9

192727

9

13 23233




Then

The Same Result as Expanding First then Integrating Term-by-Term Using the Sum Rule

023232 33

9

13313 CxxxCxxxdxx



GamePlan: Integ by Substitution

1. Choose a (clever) substitution, u = u(x), that “simplifies” the Integrand, f(x)

2. Find the Integrating Factor, dx, in terms of x and du by:

duuhduxu

dxd

dxxudx

d

dx

duxuu

1



GamePlan: Integ by Substitution

3. After finding dx = r(h(u), du) Sub Out the Integrand and Integrating Factor to arrive at an equivalent Integral of the form:

4. Evaluate the transformed integral by finding the AntiDerivative H(u) for h(u)

5. BackSub u = u(x) into H(u) to eliminate u in favor of x to obtain the x-Result:

duuhdxxf

CxuHdxxf



Example: Substitution with e

Find SOLUTION: First, note that none of the rules from the

Previous lecture on §5.1 will immediately resolve this integral

Need to choose a substitution that yields a simpler integrand with which to work• Perhaps if the radicand were simpler, the

§5.1 rules might apply

dxeexZ xx 7




Try Letting: Take d/dx of Both Sides

Solving for dx: Now from u-Definition:

Thendx →

xxx eeedx

d

dx

duu

dx

d 017

dxdue

edx

du

e

dxx

xx

1

77 ueeu xx

duu

due

dxx 7

11




Now Sub Out in original AntiDerivative:

This process yields

This works out VERY Well

Now can BackSub for u(x)

7 ue x ue x 7 duu

dx7

1

duu

uudxeexZ xx 7

17 7

Ku

duuuRxZ

23

2321




Using u(x) = e−x+7:

Thus the Final Result:

• This Result can be verified by taking the derivative dZ/dx which should yield the original integrand

K

eK

uuRxZ

x

3

72

3

223

23

KedxeexZ xxx 23

73

2 7



Example: Sub Rational Expression

Find

SOLUTION: Try: Taking du/dx

find This produces

dx

xx

xxZ

122

243 2

3123 2 122122 xxxxu

dux

udu

uxdx

24

3

24

3 2

2

duudu

x

u

u

xxZ

324

324 2




Solving

Thus the Answer

An Alternative u:

CxxCu

duuxZ 2312

2

1222

3

2

33

Cxxdxxx

xxZ

322

3 2122

2

3

122

24

24122 2 xdx

duxxu




SubOut x using:

Find

Then

• The Same Result as before

24&122 2

x

dudxxxu

duudu

xu

xdx

xx

xxZ

31

33 2 24

124

122

24

Cxxuu

duuxZ 1222

3

2

3

32232

3231



Example DE Model for Annuities

Li Mei is a Government Worker who has an annuity referred to as a 403b. She deposits money continuously into the 403b at a rate of $40,000 per year, and it earns 2.6% annual interest.

Write a differential equation modeling the growth rate of the net worth of the annuity, solve it, and determine how much the annuity is worth at the end of 10 years.




SOLUTION: TRANSLATE: The 403b has two ways

in which it grows yearly: • The annual Deposit by Li Mei = $40k • The annual interest accrued = 0.026·A

– Where A is the current Amount in the 403b

Then the yearly Rate of Change for the Amount in the 403b account

k40$ $k/yrin 026.0 A




This DE is Variable Separable

Affecting the Separation and Integrating

Find the AntiDerivative by Substitution Let: Then:

dtdAA026.040

1

Au 026.040

dudAdA

du

026.0

1026.00




SubOut A in favor of u:

Integrating:1

1

026.0

1Ctdu

u

21026.0&026.0ln CCCCtu




Note that u = $40k + 0.026A is always positive, so the ABS-bars can be dispensed with

Now BackSub Solve for A(t) by

raising e to the power of both sidesFind the General(Includes C) solution:

CtAu 026.0026.040lnln

CtA ee 026.0026.040ln




Use the KNOWN data that at year-Zero there is NO money in the 403b; i.e.; (t0,A0) = (0,A(0)) = (0,0)

Sub (0,0) into the General Soln to find C

Or Thus the

particular soln

40ln40

026.0

400

0026.0

CCC

eee

689.340lnln CeC

026.0

40

026.0

40 689.3026.040ln026.0

tt eeA




Using the Log property Find

Factoring Out the 40

Then at 10 years the 403b Amount

vuvu aaa

026.0

4040

026.0

40

026.0

40 026.0026.040ln40ln026.0

ttt eeeeA

026.0

140

026.0

teA

k816.456$

026.0

2969.0k40$

026.0

140

10026.0

e

A



WhiteBoard Work

Problems From §5.2• P61 → Retirement Income vs. Outcome• P66 → Price Sensitivity to Supply &

Demand

http://www.google.com/url?sa=i&rct=j&q=retirement+income+outgo&source=images&cd=&cad=rja&docid=ECBGV9ctFcaVnM&tbnid=Dpxu8G9eVr5YjM:&ved=0CAUQjRw&url=http://incomeandoutgo.blogspot.com/2010_11_01_archive.html&ei=4hbvUZn7IaqoigLTsoGADg&bvm=bv.49641647,d.cGE&psig=AFQjCNGLPiFMX5OmdY7SISSs0NfhjmPYAQ&ust=1374709649471006



All Done for Today

SubstitutionCity

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Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

Appendix

–

srsrsr 22



ConCavity Sign Chart

a b c

−−−−−−++++++ −−−−−− ++++++

x

ConCavityForm

d2f/dx2 Sign

Critical (Break)Points Inflection NO

InflectionInflection



http://www.google.com/url?sa=i&rct=j&q=retirement%20income&source=images&cd=&cad=rja&docid=oM696xiRoyGnlM&tbnid=aY1RyvFYgF5IQM:&ved=0CAUQjRw&url=http://www.theinsurancesuite.com/Retirement_Income.html&ei=FjrvUfvBOIreiAK9roHoCA&bvm=bv.49641647,d.cGE&psig=AFQjCNF9Ne0wmNbc91e5uKaSSW89EaNX9g&ust=1374718814904352





















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