sec 7.5: strategy for integration integration is more challenging than differentiation. no hard and...
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Sec 7.5: 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.
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Sec 7.5: STRATEGY FOR INTEGRATION
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, rational in sine & cos,
1)Try subsitution 2)Try parts 3)Manipulate integrand4)Relate to previous Problems 5)Use several methods
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Sec 7.5: STRATEGY FOR INTEGRATION
4-step strategy
1 Simplify the Integrand if Possible
![Page 4: Sec 7.5: STRATEGY FOR INTEGRATION integration is more challenging than differentiation. No hard and fast rules can be given as to which method applies](https://reader035.vdocuments.us/reader035/viewer/2022062804/5697bf8b1a28abf838c8ae8f/html5/thumbnails/4.jpg)
Sec 7.5: STRATEGY FOR INTEGRATION
4-step strategy
2 Look for an Obvious Substitutionfunction and its derivative
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Sec 7.5: STRATEGY FOR INTEGRATION
4-step strategy
3 Classify the integrand according to Its formTrig fns, rational fns, by parts, radicals, rational in sine & cos,
7.2 7.4 7.1 7.3 7.4
4 Try Again
1)Try subsitution 2)Try parts 3)Manipulate integrand4)Relate to previous Problems 5)Use several methods
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Sec 7.5: STRATEGY FOR INTEGRATION
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Sec 7.5: 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
7.33 Integrand = )()( xgxf
poly
We know how to integrate all the way
by parts (many times)
5 Integrand contains: only trig
7.2 6 Integrand = rationalPartFrac
f & f’
7 Integrand = rational in sin & cos
Convert into rational )tan(2
xt
8 Back to original 2-times by part original
xdxex sin xdxex cos
9 Combination:
dxexx x34 )3(
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Sec 7.5: STRATEGY FOR INTEGRATION
Trig fns
Partial fraction
by parts
radicals
rational sine&cos
Subsit or combination
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Sec 7.5: STRATEGY FOR INTEGRATION
Trig fns
Partial fraction
by parts
radicals
rational sine&cos
Subsit or combination
![Page 10: Sec 7.5: STRATEGY FOR INTEGRATION integration is more challenging than differentiation. No hard and fast rules can be given as to which method applies](https://reader035.vdocuments.us/reader035/viewer/2022062804/5697bf8b1a28abf838c8ae8f/html5/thumbnails/10.jpg)
Sec 7.5: STRATEGY FOR INTEGRATION
Trig fns
Partial fraction
by parts
radicals
rational sine&cos
Subsit or combination
(Substitution then combination)
![Page 11: Sec 7.5: STRATEGY FOR INTEGRATION integration is more challenging than differentiation. No hard and fast rules can be given as to which method applies](https://reader035.vdocuments.us/reader035/viewer/2022062804/5697bf8b1a28abf838c8ae8f/html5/thumbnails/11.jpg)
Sec 7.5: STRATEGY FOR INTEGRATION
Trig fns
Partial fraction
by parts
radicals
rational sine&cos
Subsit or combination
(Substitution then combination)
![Page 12: Sec 7.5: STRATEGY FOR INTEGRATION integration is more challenging than differentiation. No hard and fast rules can be given as to which method applies](https://reader035.vdocuments.us/reader035/viewer/2022062804/5697bf8b1a28abf838c8ae8f/html5/thumbnails/12.jpg)
Sec 7.5: STRATEGY FOR INTEGRATION
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
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Sec 7.5: 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
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Sec 7.5: 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?
If g(x) elementary
g’(x) elementary
FACT:
need not be an elementary
If f(x) elementary
NO:
x
adttfxF )()(
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Sec 7.5: STRATEGY FOR INTEGRATION
CAN WE INTEGRATE ALL CONTINUOUS FUNCTIONS?
Will our strategy for integration enable us to find the integral of every continuous function?
If g(x) elementary
g’(x) elementary
FACT:
need not be an elementary
If f(x) elementary
NO:
x
adttfxF )()(
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.