ch8 : strategy for integration
DESCRIPTION
Ch8 : STRATEGY FOR INTEGRATION. integration is more challenging than differentiation. No hard and fast rules can be given as to which method applies in a given situation, but we give some advice on strategy that you may find useful. - PowerPoint PPT PresentationTRANSCRIPT
Ch8: STRATEGY FOR INTEGRATION
integration is more challenging than differentiation.
No hard and fast rules can be given as to which method applies in a given situation, but we give some advice on strategy that you may find useful.
how to attack a given integral, you might try the following four-step strategy.
4-step strategy
1 Simplify the Integrand if Possible
2 Look for an Obvious Substitution
3 Classify the Integrand According to Its Form
4 Try Again
function and its derivative
Trig fns, rational fns, by parts, radicals,
1)Try subsitution 2)Try parts 3)Manipulate integrand4)Relate to previous Problems 5)Use several methods
Ch8: STRATEGY FOR INTEGRATION
4-step strategy
1 Simplify the Integrand if Possible
Ch8: STRATEGY FOR INTEGRATION
4-step strategy
2 Look for an Obvious Substitutionfunction and its derivative
Ch8: STRATEGY FOR INTEGRATION
xdxsin
xdxcos
xdx2sec
xdx2csc
xdxx tansec
xdxx cotcsc
x
dx
12x
dx
dxex
4-step strategy
3 Classify the integrand according to Its formTrig fns, rational fns, by parts, radicals,
8.2 8.4 8.1 8.3
4 Try Again
1)Try subsitution 2)Try parts 3)Manipulate integrand4)Relate to previous Problems 5)Use several methods
Ch8: STRATEGY FOR INTEGRATION
3 Classify the integrand according to Its form
1 Integrand contains: xln
by partsln and its derivative
2 Integrand contains:11 sin,tan
by partsf and its derivative
4 Integrand radicals:2222 , axxa
8.33 Integrand = )()( xgxf
poly
We know how to integrate all the way
by parts (many times)
5 Integrand contains: only trig
8.2
6 Integrand = rationalPartFrac
f & f’
7 Back to original 2-times by part original
xdxex sin xdxex cos
8 Combination:
dxexx x34 )3(
Ch8: STRATEGY FOR INTEGRATION
xdx3sec
xdxx mn cossin
xdxx mn tansec
xdxxf cos)(sin
xdxxf 2sec)(tan
Ch8: STRATEGY FOR INTEGRATION
dxexx x)1(
xdxx 1tan
dxx
x8
5
cos
sin 2/32 )4( x
dx
dx
x
x
1
22
5
13x
dx
dxx
x4cos1
2sin
xdxe x 2sin
dxx
x2sin9
cos
xx
dx
1 xdxx 2cot2csc 36
dxxx
x
)1)(1(
22
2/32
2
)4( x
dxx
dxxx )2cos()3sin(
122 111
dxx 2)tan2(
dxxx )2sin(2
dxxx 3coscos
42xx
dx
dxexx xsin)cos2)2(sin(
dx
x
x
1csc
sin2
16 xx
dx
22
1523 xxx
dx
dxx )sin(ln 2
112
2
2
41 x
dxx
102
Trig fns
Partial fraction
by parts
subs Trig subs
combinationPower of Obvious subsothers
Bac
ko
rig
inal
seve
ral
Trig fns
Partial fraction
by parts
subs
Ch8: STRATEGY FOR INTEGRATION
dxexx x)1(
xdxx 1tan
dxx
x8
5
cos
sin
2/32 )4( x
dx
dx
x
x
1
22
5
13x
dx
dxx
x4cos1
2sin
xdxe x 2sin
dxx
x2sin9
cos xx
dx
1 xdxx 2cot2csc 36
dxxx
x
)1)(1(
22 2/32
2
)4( x
dxx
dxxx )2cos()3sin(
122 111
dxx 2)tan2( dxxx )2sin(2
dxxx 3coscos
42xx
dx
dxexx xsin)cos2)2(sin(
dx
x
x
1csc
sin2 16 xx
dx
22
1523 xxx
dx
dxx )sin(ln 2
112
2
2
41 x
dxx
102
Trig subs
combinationPower of Obvious subsothers
Bac
ko
rig
inal
seve
ral
Ch8: STRATEGY FOR INTEGRATION132131
Trig fns
Partial fraction
by parts
Subs Trig subs
combinationPower of Obvious subsothers
Bac
ko
rig
inal
seve
ral
Trig fns
Partial fraction
by parts
Subs
Ch8: STRATEGY FOR INTEGRATION132131
Trig subs
combinationPower of Obvious subsothers
Bac
ko
rig
inal
seve
ral
Ch8: STRATEGY FOR INTEGRATION
Trig fns
Partial fraction
by parts
Subs Trig subs
combinationPower of Obvious subsothersB
ack
ori
gin
alse
vera
l
Trig fns
Partial fraction
by parts
Subs
Ch8: STRATEGY FOR INTEGRATION
Trig subs
combinationPower of Obvious subsothersB
ack
ori
gin
alse
vera
l
(Substitution then combination)Ch8: STRATEGY FOR INTEGRATION
Trig fns
Partial fraction
by parts
Subs Trig subs
combination
Bac
ko
rig
inal
seve
ral
Trig fns
Partial fraction
by parts
Subs
(Substitution then combination)Ch8: STRATEGY FOR INTEGRATION
Trig subs
combination
Bac
ko
rig
inal
seve
ral
elementary functions.
polynomials, rational functionspower functionsExponential functions logarithmic functions
trigonometric inverse trigonometrichyperbolic inverse hyperbolic
all functions that obtained from above by 5-operations , , , ,
Ch8: STRATEGY FOR INTEGRATION
If g(x) elementary
FACT:
need not be an elementary
If f(x) elementary
NO:
x
adttfxF )()(
g’(x) elementary
CAN WE INTEGRATE ALL CONTINUOUS FUNCTIONS?
Will our strategy for integration enable us to find the integral of every continuous function?
YES or NO
YES or NO
)(xf Continuous.if Anti-derivative )(xF exist?
dxex2
Ch8: STRATEGY FOR INTEGRATION
CAN WE INTEGRATE ALL CONTINUOUS FUNCTIONS?
elementary functions.
polynomials, rational functionspower functionsExponential functions logarithmic functions
trigonometric inverse trigonometrichyperbolic inverse hyperbolic
all functions that obtained from above by 5-operations , , , ,
Will our strategy for integration enable us to find the integral of every continuous function?
YES
NO
dxex2
Ch8: STRATEGY FOR INTEGRATION
has an antiderivative2
)( xexf x
a
t dtexF2
)(
This means that no matter how hard we try, we will never succeed in evaluating in terms of the functions we know.
is not an elementary.
In fact, the majority of elementary functions don’t have elementary antiderivatives.
Ch8: STRATEGY FOR INTEGRATION
If g(x) elementary
FACT:
need not be an elementary
If f(x) elementary
NO:
x
adttfxF )()(
g’(x) elementary
Example dxx cos 1 Example dxx sec 1
xy 1sec
ydyxxdxx secsec sec 11
Cyyxx tanseclnsec 1
Cxxxx )tan(seclnsec 11
Cxxxx 1lnsec 21
Example dxx cos 1
dxxxxxdxxdx
d coscos cos 111
dx
xxxx
1
1cos
2
1
dxxxx
x cos21
1
dxxxx
x )(cos21
2
2
11
Cxxx 21 1cos
Ch8: STRATEGY FOR INTEGRATION