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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.

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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 integrand 4)Relate to previous Problems 5)Use several methods Ch8: STRATEGY FOR INTEGRATION

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4-step strategy 1 Simplify the Integrand if Possible Ch8: STRATEGY FOR INTEGRATION

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4-step strategy 2 Look for an Obvious Substitution function and its derivative Ch8: STRATEGY FOR INTEGRATION

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4-step strategy 3 Classify the integrand according to Its form Trig fns, rational fns, by parts, radicals, 8.28.48.18.3 4 Try Again 1)Try subsitution 2)Try parts 3)Manipulate integrand 4)Relate to previous Problems 5)Use several methods Ch8: STRATEGY FOR INTEGRATION

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3 Classify the integrand according to Its form 1 Integrand contains: by parts ln and its derivative 2 Integrand contains: by parts f and its derivative 4 Integrand radicals : 8.3 3 Integrand = We know how to integrate all the way by parts (many times) 5 Integrand contains: only trig 8.2 6 Integrand = rational PartFrac f & f’ 7 Back to original 2-times by part original 8 Combination: Ch8: STRATEGY FOR INTEGRATION

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122111 112 102 Trig fns Partial fraction by parts subs Trig subs combination Power of Obvious subs others Back original several

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Trig fns Partial fraction by parts subs Ch8: STRATEGY FOR INTEGRATION 122111 112 102 Trig subs combination Power of Obvious subs others Back original several

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Ch8: STRATEGY FOR INTEGRATION 132 131 Trig fns Partial fraction by parts SubsTrig subs combination Power of Obvious subs others Back original several

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Trig fns Partial fraction by parts Subs Ch8: STRATEGY FOR INTEGRATION 132 131 Trig subs combination Power of Obvious subs others Back original several

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Ch8: STRATEGY FOR INTEGRATION Trig fns Partial fraction by parts SubsTrig subs combination Power of Obvious subs others Back original several

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Trig fns Partial fraction by parts Subs Ch8: STRATEGY FOR INTEGRATION Trig subs combination Power of Obvious subs others Back original several

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(Substitution then combination) Ch8: STRATEGY FOR INTEGRATION Trig fns Partial fraction by parts SubsTrig subs combination Back original several

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Trig fns Partial fraction by parts Subs (Substitution then combination) Ch8: STRATEGY FOR INTEGRATION Trig subs combination Back original several

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elementary functions. polynomials, rational functions power functions Exponential functions logarithmic functions trigonometric inverse trigonometric hyperbolic 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: g’(x) elementary

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CAN WE INTEGRATE ALL CONTINUOUS FUNCTIONS? Will our strategy for integration enable us to find the integral of every continuous function? YES or NO Continuous.ifAnti-derivativeexist? Ch8: STRATEGY FOR INTEGRATION

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CAN WE INTEGRATE ALL CONTINUOUS FUNCTIONS? elementary functions. polynomials, rational functions power functions Exponential functions logarithmic functions trigonometric inverse trigonometric hyperbolic 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 Ch8: STRATEGY FOR INTEGRATION

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has an antiderivative 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: g’(x) elementary

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Example

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Ch8: STRATEGY FOR INTEGRATION

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