Fi 1025

Notation If F x fly ice if Fay is an antiderivative of fadthen we write fix d Fix C We callthis an indefiniteintegral

In contrast fabfly dx is a definite integral

Practice with antiderivatives indefinite integrals

I 1 dx Xtc

X dx I x'te

f x'dx s Htc

Generator fxndx XI C n I

Tx dx x dx c Zz x c 3 t C

I dx f E dx In x C Note Eo is undefined

I ex dx exec Je teeth C

f Sin x dx cos x C f sin kx dx th costextC

f cos x dx sin x t C cos lex DX thesink x C

Practice with definite integrals

x 6 26021 6,1

30 2 2 3 l30.102 2.103 30.02 2.03

3000 2000

I 3rx dx L 3xkdx 37 I16

2 42 4 2 If 21563 2543 2.64 2.8 1120

3 sin x dx 3 cos x IFf3cos2x 3Coso 1

3 l t 3 I

If 1 1 dx 16 seegrapharea.se raea8oIf xdxtfo x dx

Eli Elfo 8 t 8 0 160

Mon 1028

Application Average value of a function

Motivating example through 8 gamesClemson 8 winsOklahoma 7 wins

USC 3 wins

The average wins 8 7332 6 wins

Now consider a continuous quantity fixThe average value of fix on a Exeb is Jabfu dx

x1 Suppose the velocity of a car is given by vCH 30T

for O E te 2 Find its average velocity

Ant Toto 30 t dt I 3 31UH

513 02 5 23 5.0 40y

Suppose the elevation on a 5 mile hike is

f x 60 3 650 2 1200 4500 ft IFind the average elevation

11 2000

0 60 3 650 41200 4500DX

s 15 4 6 600 x't 4500x to

t 15 5 EF 53 600.5 14500 5 3958.3ft

2erivatiresrsintegrtsJervatives have formulas product quotient chain rules

So we can always take the derivative of e.g ekin x cos x

Integrals have no such formula

For example I e x dy has no closed form

Computing integrals is more of an ant

But there are some rules tricks we'll lean on now

Usubstitution like the chain rule in reverse

Recall the chain rule Fegan ffgcxD.glx where F

Now integrate both sides F glx 1C I f glad glad DXNow let a gCH du du

off g'M F u tCJfCuTdu g x dx

Examples

I 212 1 dx let u 2x tl off L du 2 dx

f u da ITU C I 2 1 C

f3ew d let u 10x off _to du look O

feu fo da fodu 3 dx

Foe C foetch't C

1 41 5 619 dy let u X5t6 off 5x du 5x dy

u9 Edu Edu x4dx

t Ufo 1C foll 1C Fo x5 6 1Ced 1030

floos xp Sindy let a CosX du sin dx

Just du UI C sc

fYdf let u Hx2 du 2x dx Edu xd x

Iii't du Y C Uktc txt C

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