Transcript Sec 7.5
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 function and its derivative 3 Classify the Integrand According to Its Form Trig fns, rational fns, by parts, radicals, rational in sine & cos, 4 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 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, 7.2 4 7.4 7.1 7.3 7.4 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 Sec 7.5: STRATEGY FOR INTEGRATION 3 1 Classify the integrand according to Its form Integrand contains: ln and its derivative ln x by parts 2 Integrand contains: tan 1 , sin 1 f and its derivative by parts 4 Integrand radicals: a2 x2 , x2 a2 3 7.3 5 Integrand = f ( x) g ( x) by parts (many times) poly (x Integrand contains: only trig 6 7.2 We know how to integrate all the way 4 3 x)e3 x dx Integrand = rational PartFrac f & f’ 8 Back to original x e sin xdx 9 2-times by part original e x cos xdx 7 Integrand = rational in sin & cos Convert into rational Combination: t tan( 2x ) Sec 7.5: STRATEGY FOR INTEGRATION Partial fraction Trig fns Subsit or combination by parts radicals rational sine&cos Sec 7.5: STRATEGY FOR INTEGRATION Partial fraction Trig fns Subsit or combination by parts radicals rational sine&cos Sec 7.5: STRATEGY FOR INTEGRATION Partial fraction Trig fns Subsit or combination by parts (Substitution then combination) radicals rational sine&cos Sec 7.5: STRATEGY FOR INTEGRATION Partial fraction Trig fns Subsit or combination by parts (Substitution then combination) radicals rational sine&cos Sec 7.5: STRATEGY FOR INTEGRATION CAN WE INTEGRATE ALL CONTINUOUS FUNCTIONS? if f (x ) YES or NO Continuous. Will our strategy for integration enable us to find the integral of every continuous function? Anti-derivative YES or NO F (x ) e x2 exist? dx Sec 7.5: STRATEGY FOR INTEGRATION elementary functions. polynomials, trigonometric rational functions inverse trigonometric power functions hyperbolic Exponential functions inverse hyperbolic logarithmic functions all functions that obtained from above by 5-operations , , , , CAN WE INTEGRATE ALL CONTINUOUS FUNCTIONS? Will our strategy for integration enable us to find the integral of every continuous function? e x2 dx NO YES Sec 7.5: STRATEGY FOR INTEGRATION elementary functions. FACT: polynomials, trigonometric rational functions inverse trigonometric power functions hyperbolic Exponential functions inverse hyperbolic logarithmic functions all functions that obtained from above by 5-operations If g(x) elementary , , , , g’(x) elementary NO: If f(x) elementary CAN WE INTEGRATE ALL CONTINUOUS FUNCTIONS? x Will our strategy for integration enable us to find the integral of every continuous function? F ( x) f (t )dt a need not be an elementary 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? FACT: NO: If g(x) elementary If f(x) elementary g’(x) elementary x F ( x) f (t )dt a need not be an elementary f ( x) e x2 has an antiderivative x F ( x) e dt a t2 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.