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INTEGRATION

# INTEGRATION - Strategy for Integration As we have seen...

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Strategy for Integration As we have seen, integration is more challenging than differentiation. In finding the deriv- ative of a function it is obvious which differentiation formula we should apply. But it may not be obvious which technique we should use to integrate a given function. 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 5.6, and partial fractions in Exercises 5.7 and Appendix G. But in this section we present a collection of miscellaneous integrals in random order and the main challenge is to recognize which technique or formula to use. 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. A prerequisite for strategy selection is a knowledge of the basic integration formulas. In the following table we have collected the integrals from our previous list together with several additional formulas that we have learned in this chapter. Most of them should be memorized. It is useful to know them all, but the ones marked with an asterisk need not be memorized since they are easily derived. Formula 19 can be avoided by using partial frac- tions, and trigonometric substitutions can be used in place of Formula 20. Table of Integration Formulas Constants of integration have been omitted. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 1 7. 18. *19. *20. y dx s x 2 a 2 ln x s x 2 a 2 y dx x 2 a 2 1 2 a ln x a x a y dx s a 2 x 2 sin 1 x a y dx x 2 a 2 1 a tan 1 x a y cosh x dx sinh x y sinh x dx cosh x y cot x dx ln sin x y tan x dx ln sec x y csc x dx ln csc x cot x y sec x dx ln sec x tan x y csc x cot x dx csc x y sec x tan x dx sec x y csc 2 x dx cot x y sec 2 x dx tan x y cos x dx sin x y sin x dx cos x y a x dx a x ln a y e x dx e x y 1 x dx ln x n 1 y x n dx x n 1 n 1 1

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Once you are armed with these basic integration formulas, if you don’t immediately see how to attack a given integral, you might try the following four-step strategy. 1. Simplify the Integrand if Possible Sometimes the use of algebraic manipulation or trigonometric identities will simplify the integrand and make the method of integration obvious. Here are some examples: 2. Look for an Obvious Substitution Try to find some function in the inte- grand whose differential also occurs, apart from a constant factor. For instance, in the integral we notice that if , then . Therefore, we use the substitu- tion instead of the method of partial fractions. 3. Classify the Integrand According to Its Form If Steps 1 and 2 have not led to the solution, then we take a look at the form of the integrand . (a) Trigonometric functions. If is a product of powers of and , of and , or of and , then we use the substitutions recom- mended in Section 5.7 and Additional Topics: Trigonometric Integrals .
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INTEGRATION - Strategy for Integration As we have seen...

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