CALCULUS 2: STRATEGY FOR EVALUATING INTEGRALS1.A few general tacticsIn 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 antiderivativeF(x)for the function involved and deal later with the constant of integration in an indefiniteintegral and the limits of a definite integral.However when usingsubstitutions on adefinite integral, you can avoid substituting back to the original variable by insteadchanging the limits of integration to the corresponding values for the new variable.Start with the easiest possibilities:Several techniques might apply to one integral, sostart try the easiest first (recognizing a previously known integral, from memory or tables)and work through to the more sophisticated options (like inverse trigonometric substitutionsand integration by parts.)Repeat as necessary:All methods except recognizing a known integral give one or severalnew simpler integrals, so the process needs to be applied repeatedly until every part of theanswer is found by reducing to a previously known integral.2.A detailed strategyHere is one detailed approach to evaluating integrals. It is based on the discussion in Section 7.5 andcovers all the methods we have seen in class. For practice and test review, I strongly recommendselecting exercises from Section 7.5, so that you must decide what methods to use as well as thenapplying 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 anintegral as one you have already worked out how to evaluate.The corollary of this is that youshould memorize the most common integrals, and collect the ones that you encounter often buthave not memorized (yet) on aformula sheet.The formulas should be as flexible as possible withadjustable constants to avoid routine substitutions: notcosx dx= sinx+Cbutcosax dx=1asinax+C.B. Do basic simplifications.Simplify firstis 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 upsums and differencesinto separate in-tegrals, takingconstant factorsout in front of each integral andrewriting rootsas fractionalpowers.It is also often useful to eliminate divisions by rewriting powers in the denominator as negativepowers in the numerator, and using trig identities like converting a factor cosxin the denominatorinto a factor secxin the numerator.For example,x27-1sin 2x+5x3√1 +x2dxcould be rewritten as17x2dx-csc 2x dx+ 5x(1 +x2)-1/3dxDate: February 25, 2003.
2CALCULUS 2: STRATEGY FOR EVALUATING INTEGRALSOne important special situation is integrals ofproducts of powers of trigonometric functions,which will be discussed below.