Integration by substitution

Recall CalledCalled indefinite integral

c4 a doc Most general anti derivative of 1 sa

On open F Cal t Cuterval A Constant

Particular ants derivative F Cal 1 Cx

why it matters Detinite integral Net Area

FTO C 7 Cx doe Fcb Fca

Aim Find anti derivatives for as many functions as possible

Cove Examples Rulesrt I

Jardic On 0,0 ou C N o

Joe doc In 1 1 C

tacosb c

b t

J b da

S Sinixt doc cos x C Joss x da Siu x

J da J da arcsiu x arecas a t

J doc arctanca t C constant multipleSum rule in reverse rule in reverse

tffcxi gcxldx ffcxldx fgcxsdx.SC x1dx cftcx1dx

Integration by Substitution Chain Race in reverse

F'Cal 7 x

chain RuleI

dd.cfcgc.ci F Cgc g Ge f gcses g x

JHgcxngic.ci doc F gcn C

ExampleY J cos oil 2x dx

1 x cos x g x of 7 goal g x _cos oil 2x

JC's 74 doc sin x C hence let FCK sin x

7Cgca g1Cx F gcn

Jess sa z c da Sin Xi C

z g E s next doc

Out byconstantf x e g Cal cos x

careful i g Coal 2 Sinixt e nextL

fcgcx.gg G

Je du e t C hence let FCal e

I I

J e sines doc E fight g Cal da z FCgla 1s x

e t C

Y g h doc

Problem Not obviously of the form 7cg x g la

Clever Observation day Tulsa and14 tubes

Let 7cal se and g Cx Lucah Figlang by

Jac du sir hence let Fcx 4 2

J1u da tuba t C

m For this to he must

spot 7cal and god saw that integrand is of

Form 7cgCx g CoeC Could be very

hard

4 Be able to integrate find antiderivative of 7 x

MovesjApproah Integration by Substitution

Examine Integrand Is it of the form

f gcn g coal up to multiplication by a constant

Example

1 x e 2 cos Gce sin ex 7cg call g lat

g th Zens x

2 Introduce a new variable u and set u gcxWe're going to change the variable to u in the

indefinite integral This is done in 2 stages

a Replace dx with expression in du Justclevernotation

duug Cx DI

g ex docdag Lx

Exampledu du

U 2 cos x a 25in x dx25in x

few sin doc fe sd

f ecense

Z sidu

b Rewrite integrand in a variable

Example u 2 s x

J e du J e du

31 Calculate integral in n variable

Example

f e du Je du e t C

4g Replace u with gcse to express final answer

in X variable

Examplee C e c

Overview of Integration by substitution

u genex would u would

Cori ask II jc.clda D New variable

y Gc4 3

Replace CalculateY u with gene integral

Jtcged g'CH du f gas Fcu c Jtculdu

Replace aReplacegudda

J with a

ftcgcxhgfldg.fi J7CgcxDdu

12 The strength of this approach is that

we only really need to spot goes to

implement it

Exampley costal Siu Cai

f tancxidx Y

tan xsinc 1 sin x

cos Cod

et u cos Cal DI sinise doc DIda Siu x

2 a

I doc J I ducos x Sin la

J du 7b

3

I du J du lulul C

J tan Lx dx tu cos Cx t

4 Ju VI da

Problem No obvious choice at g caX u iduLeap of faith a x 11 I du dx

Ju VI da Jay du Jer 1 Tu du

guk u k du g u z 3 uk c

3 x Fk Z x 111 C

Debut F'coal taxi

17cg Cx g ex da Fcgcx C

7cg Cx glenda Fcgcb FcgcagCb C Not same in

Jfcu du U world

g ca

General Advice To calculate a detinitecitegral

always calculate the indefinite integral SeparatelyFirst

Example

Tia

J tanh da 2 Done above

Jtancocidx Tul six Ith TI4J tanh doc Tul cosGulfO

O

C lulus Hall C lulus oilint

Exception to Rule If we can only calculateg

J Hulda geometricallyg ca

z

Example µ za da 2

Hem No obvious choice A g ca

Leap at Faith u air 2x da L2x

can't find

V za da dintimiut

ut

top A semicircleFe Laden's 2 center o

Jo 4 za da Area 4 T E T