integration is more challenging than differentiation. 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.

1 2 3 4 4-step strategy Sec 7.5 : STRATEGY FOR INTEGRATION 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

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

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

3 4 4-step strategy Sec 7.5 : STRATEGY FOR INTEGRATION Classify the integrand according to Its form Trig fns, rational fns, by parts, radicals, rational in sine and consine 7.2 7.4 7.1 7.3 7.4 4 Try Again 1)Try subsitution 2)Try parts 3)Manipulate integrand 4)Relate to previous Problems 5)Use several methods

3 1 2 3 4 6 7 5 8 9 Sec 7.5 : STRATEGY FOR INTEGRATION Classify the integrand according to Its form 1 2 Integrand contains: Integrand contains: ln and its derivative f and its derivative by parts by parts We know how to integrate all the way 3 4 Integrand = Integrand radicals: by parts (many times) 7.3 PartFrac 6 Integrand = rational Radicals with different indexes f & f’ 7 Back to original 2-times by part  original 5 Integrand contains: only trig 7.2 8 9 Rational sine & cosine: Combination:

Sec 7.5 : STRATEGY FOR INTEGRATION 132 141 121 Partial fraction Simplify integrand Radicals(Trig subs) f & f’ Rat in sin,cos conjugate radical with diff index Trig fns by parts Power of Others(identity) Obvious subs several original Back

Sec 7.5 : STRATEGY FOR INTEGRATION 132 141 121 Partial fraction Simplify integrand Radicals(Trig subs) f & f’ Rat in sin,cos conjugate radical with diff index Trig fns by parts Power of Others(identity) Obvious subs several original Back

Sec 7.5 : 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

Sec 7.5 : 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

Sec 7.5 : 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

Sec 7.5 : 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.

Sec 7.5 : STRATEGY FOR INTEGRATION Example Let Evaluate the following integrals in terms of F

Sec 7.5 : STRATEGY FOR INTEGRATION

Sec 7.5 : STRATEGY FOR INTEGRATION 102 122 111 112 Partial fraction Simplify integrand Radicals(Trig subs) f & f’ conjugate radical with diff index Trig fns by parts Power of Others(identity) Obvious subs Rational in sin & cos several original Back

Sec 7.5 : STRATEGY FOR INTEGRATION 102 122 111 112 Partial fraction Simplify integrand Radicals(Trig subs) f & f’ conjugate radical with diff index Trig fns by parts Power of Others(identity) Obvious subs Rational in sin & cos several original Back

Sec 7.5 : STRATEGY FOR INTEGRATION 142 141 Partial fraction Simplify integrand Radicals(Trig subs) f & f’ conjugate radical with diff index Trig fns by parts Power of Others(identity) Obvious subs Rational in sin & cos several original Back

Sec 7.5 : STRATEGY FOR INTEGRATION 142 141 Partial fraction Simplify integrand Radicals(Trig subs) f & f’ conjugate radical with diff index Trig fns by parts Power of Others(identity) Obvious subs Rational in sin & cos several original Back

Sec 7.5 : STRATEGY FOR INTEGRATION 132 131 Partial fraction Simplify integrand Radicals(Trig subs) f & f’ conjugate radical with diff index Trig fns by parts Power of Others(identity) Obvious subs Rational in sin & cos several original Back

Sec 7.5 : STRATEGY FOR INTEGRATION 132 131 Partial fraction Simplify integrand Radicals(Trig subs) f & f’ conjugate radical with diff index Trig fns by parts Power of Others(identity) Obvious subs Rational in sin & cos several original Back

Sec 7.5 : STRATEGY FOR INTEGRATION Partial fraction Simplify integrand Radicals(Trig subs) f & f’ conjugate radical with diff index Trig fns by parts Power of Others(identity) Obvious subs Rational in sin & cos several original Back

Sec 7.5 : STRATEGY FOR INTEGRATION Partial fraction Simplify integrand Radicals(Trig subs) f & f’ conjugate radical with diff index Trig fns by parts Power of Others(identity) Obvious subs Rational in sin & cos several original Back

Sec 7.5 : STRATEGY FOR INTEGRATION Partial fraction Simplify integrand others Radicals(Trig subs) f & f’ conjugate radical with diff index Trig fns by parts Power of Others(identity) Obvious subs Rational in sin & cos several original Back

Example Example

Example