Sec 7.5: STRATEGY FOR INTEGRATION integration is more challenging than differentiation. No hard and...

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.


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


4-step strategy

1 Simplify the Integrand if Possible


4-step strategy

2 Look for an Obvious Substitutionfunction and its derivative


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


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(


Trig fns

Partial fraction

by parts

radicals

rational sine&cos

Subsit or combination


Trig fns

Partial fraction

by parts

radicals

rational sine&cos

Subsit or combination

(Substitution then combination)


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



elementary functions.

polynomials, rational functionspower functionsExponential functions logarithmic functions

trigonometric inverse trigonometrichyperbolic inverse hyperbolic

all functions that obtained from above by 5-operations , , , ,


YES

NO

dxex2



elementary functions.

polynomials, rational functionspower functionsExponential functions logarithmic functions

trigonometric inverse trigonometrichyperbolic inverse hyperbolic

all functions that obtained from above by 5-operations , , , ,


If g(x) elementary

g’(x) elementary

FACT:

need not be an elementary

If f(x) elementary

NO:

x

adttfxF )()(




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.

Sec 7.5: STRATEGY FOR INTEGRATION integration is more challenging than differentiation. No hard and...

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