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Ch8 : STRATEGY FOR INTEGRATION

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.

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Ch8 : STRATEGY FOR INTEGRATION

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  1. 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.

  2. Ch8: STRATEGY FOR INTEGRATION 4-step strategy 1 Simplify the Integrand if Possible 2 Look for an Obvious Substitution function and its derivative 3 Classify the Integrand According to Its Form Trig fns, rational fns, by parts, radicals, 4 Try Again 1)Try subsitution 2)Try parts 3)Manipulate integrand 4)Relate to previous Problems 5)Use several methods

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

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

  5. Ch8: STRATEGY FOR INTEGRATION 4-step strategy 3 Classify the integrand according to Its form Trig 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 integrand 4)Relate to previous Problems 5)Use several methods

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

  7. Ch8: STRATEGY FOR INTEGRATION 122 111 102 112 subs Partial fraction Trig subs Trig fns by parts combination Power of Obvious subs others several Back original

  8. Ch8: STRATEGY FOR INTEGRATION 122 111 102 112 subs Partial fraction Trig subs Trig fns by parts combination Power of Obvious subs others several Back original

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

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

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

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

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

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

  15. Ch8: STRATEGY FOR INTEGRATION 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 FACT: If g(x) elementary g’(x) elementary NO: If f(x) elementary need not be an elementary

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

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

  18. Ch8: STRATEGY FOR INTEGRATION FACT: If g(x) elementary g’(x) elementary NO: If f(x) elementary need not be an elementary has an antiderivative is not an elementary. This means that no matter how hard we try, we will never succeed in evaluating in terms of the functions we know. In fact, the majority of elementary functions don’t have elementary antiderivatives.

  19. Example Example

  20. Example

  21. Ch8: STRATEGY FOR INTEGRATION

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