n_13535 - CALCULUS 2 STRATEGY FOR EVALUATING INTEGRALS 1 A...

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CALCULUS 2: STRATEGY FOR EVALUATING INTEGRALS 1. A few general tactics In trying to evaluate an integral with the collection of methods we have seen in Chapters 5 and 7, some general ideas are worth keeping in mind: Find antiderivatives first: It is often best to start by seeking any one antiderivative F ( x ) for the function involved and deal later with the constant of integration in an indefinite integral and the limits of a definite integral. However when using substitutions on a definite integral , you can avoid substituting back to the original variable by instead changing the limits of integration to the corresponding values for the new variable. Start with the easiest possibilities: Several techniques might apply to one integral, so start try the easiest first (recognizing a previously known integral, from memory or tables) and work through to the more sophisticated options (like inverse trigonometric substitutions and integration by parts.) Repeat as necessary: All methods except recognizing a known integral give one or several new simpler integrals, so the process needs to be applied repeatedly until every part of the answer is found by reducing to a previously known integral. 2. A detailed strategy Here is one detailed approach to evaluating integrals. It is based on the discussion in Section 7.5 and covers all the methods we have seen in class. For practice and test review, I strongly recommend selecting exercises from Section 7.5, so that you must decide what methods to use as well as then applying the methods correctly: that is the way integration is in real life. A. Use tables of integrals and known integrals. The easiest approach is to recognize an integral as one you have already worked out how to evaluate. The corollary of this is that you should memorize the most common integrals, and collect the ones that you encounter often but have not memorized (yet) on a formula sheet. The formulas should be as flexible as possible with adjustable constants to avoid routine substitutions: not cos x dx = sin x + C but cos ax dx = 1 a sin ax + C. B. Do basic simplifications. Simplify first is a good strategy in many mathematical situations: try to simplify the function involved before starting on the calculus itself. The most general basic simplifications are breaking up sums and differences into separate in- tegrals, taking constant factors out in front of each integral and rewriting roots as fractional powers. It is also often useful to eliminate divisions by rewriting powers in the denominator as negative powers in the numerator, and using trig identities like converting a factor cos x in the denominator into a factor sec x in the numerator. For example, x 2 7 - 1 sin 2 x + 5 x 3 1 + x 2 dx could be rewritten as 1 7 x 2 dx - csc 2 x dx + 5 x (1 + x 2 ) - 1 / 3 dx Date : February 25, 2003.
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2 CALCULUS 2: STRATEGY FOR EVALUATING INTEGRALS One important special situation is integrals of products of powers of trigonometric functions , which will be discussed below.
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