7 TECHNIQUES OF INTEGRATION. As we have seen, integration is more challenging than differentiation. –In finding the derivative of a function, it is obvious.

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7 TECHNIQUES OF INTEGRATION

As we have seen, integration is more challenging than differentiation. –In finding the derivative of a function, it is obvious which differentiation formula we should apply. –However, it may not be obvious which technique we should use to integrate a given function. TECHNIQUES OF INTEGRATION

Until now, individual techniques have been applied in each section. For instance, we usually used: –Substitution in Exercises 5.5 –Integration by parts in Exercises 7.1 –Partial fractions in Exercises 7.4 TECHNIQUES OF INTEGRATION

7.5 Strategy for Integration In this section, we will learn about: The techniques to evaluate miscellaneous integrals. TECHNIQUES OF INTEGRATION

TABLE OF INTEGRATION FORMULAS

–Formula 19 can be avoided by using partial fractions. –Trigonometric substitutions can be used instead of Formula 20. TABLE OF INTEGRATION FORMULAS

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.

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 integrand 4)Relate to previous Problems 5)Use several methods

Sec 7.5: STRATEGY FOR INTEGRATION 4-step strategy 1 Simplify the Integrand if Possible

Sec 7.5: STRATEGY FOR INTEGRATION 4-step strategy 2 Look for an Obvious Substitution function and its derivative

Sec 7.5: STRATEGY FOR INTEGRATION 4-step strategy 3 Classify the integrand according to Its form Trig fns, rational fns, by parts, radicals, rational in sine & cos, Try Again 1)Try subsitution 2)Try parts 3)Manipulate integrand 4)Relate to previous Problems 5)Use several methods

Sec 7.5: STRATEGY FOR INTEGRATION

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 : Integrand = We know how to integrate all the way by parts (many times) 5 Integrand contains: only trig Integrand = rational PartFrac f & f’ 7 Integrand = rational in sin & cos Convert into rational 8 Back to original 2-times by part  original 9 Combination:

Sec 7.5: STRATEGY FOR INTEGRATION Trig fns Partial fraction by parts radicals rational sine&cos Subsit or combination

Sec 7.5: STRATEGY FOR INTEGRATION Trig fns Partial fraction by parts radicals rational sine&cos Subsit or combination

Sec 7.5: STRATEGY FOR INTEGRATION Trig fns Partial fraction by parts radicals rational sine&cos Subsit or combination (Substitution then  combination)

Sec 7.5: STRATEGY FOR INTEGRATION Trig fns Partial fraction by parts radicals rational sine&cos Subsit or combination (Substitution then  combination)

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 Continuous.ifAnti-derivativeexist?

Sec 7.5: STRATEGY FOR INTEGRATION 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

Sec 7.5: STRATEGY FOR INTEGRATION 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? If g(x) elementary g’(x) elementary FACT: need not be an elementary If f(x) elementary NO:

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: 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.