Trigonometric integrals by basic methods

Transcription

1 Roberto s Notes on Integral Calculus Chapter : Integration methods Section 7 Trigonometric integrals by basic methods What you need to know already: Integrals of basic trigonometric functions. Basic trigonometric identities. Integration by substitution. Integration by parts. What you can learn here: How to use trigonometric identities and/or the methods of substitution and parts to integrate more comple trigonometric functions. We already know how to compute integrals for many basic trig functions, such as y sin, y cos, y sec and y, the last one by using substitution. By using familiar trigonometric identities and the methods of substitution or parts, it is possible to compute integrals of more comple functions that involve trig components. Once again, rather than providing a long list of rules, here are some eamples from which we can etract a general operating principle. The integrand is not the derivative of something we know, but it is part of a basic identity, namely: sec 1 We can therefore change our integral to: sec 1 And now it is easy, since we know that y sec is the derivative of y. Therefore we can conclude that: sec 1 d c Strategy for integrating trigonometric functions If, by using a basic identity, the given integrand can be changed to one whose antiderivative we know, apply such identity and integrate as possible. Integral Calculus Chapter : Integration methods Section 7: Trigonometric integrals by basic methods Page 1

2 cos sin 1 sin sin, so we u sin, du sin cos to get: cos sin cos sin 1 1 d du du 1 sin 1 u sin cos 1 u Here we notice that the numerator is (almost) the derivative of try the substitution This is now a basic integral, so we conclude that: cos sin sin 1 1 u c sin c Strategy for integrating functions with trigonometric components When a suitable substitution can be applied to change a trigonometric integral to a simpler, algebraic one, use it. Remember that sometimes the first substitution one tries may not work, but there may be others available. Here is an eample. sec Here the obvious candidate for a substitution is u, since it is inside a fifth power. But if you try that, it will not work: check it yourself to learn why! However, we can write the integrand as: sec sec sec sec sec 1 d This suggests using the substitution u sec, du sec, which provides: sec 1 u u 1 u du u u du u c sec sec sec c Not eactly an epected conclusion, eh? Sometimes we can construct a general strategy for a special class of functions. Here is a typical case. cos For an integral like this we use the presence of an odd power. Why is this useful? Because we can separate one power of the cosine and write the remaining even power as a power of cos : cos cos cos cos cos At this point we can use the Pythagorean identity and write: cos cos 1 sin cos Since the derivative of sin is cos, the substitution u sin, du cos is effective: 1 sin cos 1 1 And the rest is easy: u du u u du Integral Calculus Chapter : Integration methods Section 7: Trigonometric integrals by basic methods Page

3 1 1 1 sin sin sin c u u du u u u c Strategy for integrating cos or sin n1 n1 When the integrand is an odd power of sine or cosine: 1. separate one power,. change the remaining even power by using the basic Pythagorean identity,. Use a substitution by letting u be the co-function of the original one. So that: 1 1 f ' u, g u f u, g ' 1 u 1 1 u u udu u u du 1 u 1 u u u du u u 1 du 1u 1u 1 1 u u u u c 1 1 c Notice that we start with integration by parts, with a single factor, but I will leave that for your practice. Knot on your finger If an integral involving a trigonometric function seems suitable for integration by parts, try it! And, of course, integration by parts can be used as well: 1 We can start by trying a substitution: 1 u, du d udu d 1 1 u udu This is now a clear candidate for integration by parts, so we let: Of course there are many more options for the use of substitutions and identities, but I leave them to your practice and eploration. Integral Calculus Chapter : Integration methods Section 7: Trigonometric integrals by basic methods Page

4 Summary When possible, use appropriate basic trigonometric identities and the method of substitution to compute integrals for trigonometric functions. Use the proper identities and use them properly! Common errors to avoid Learning questions for Section I -7 Review questions: 1. Describe two strategies for integrating trigonometric functions by using basic methods. Memory questions: f 1. Which identity is used to integrate n?. How do we re-write the integrand in order to evaluate 1 cos? Computation questions: Compute the indefinite integrals of questions 1- by using substitutions, parts and/or identities as needed. 1. sin.. cot csc Integral Calculus Chapter : Integration methods Section 7: Trigonometric integrals by basic methods Page.. cos sin cos.

5 . 6. sin e cos sec.. 7. sec sec e 10. sec 11. sec. 1. sec sec sec sec. d sec 17. sec cot csc cot cos csc cos sin ln( ). d sin cos f e sin e.. Determine the general antiderivative of the function. Use integration by parts to evaluate the integral sec. You may also want to check your answer by using the method of substitution as well.. Determine a function y f whose derivative is f ' whose graph contains the point,. and Theory questions: 1. Is trigonometric integrals a method of integration?. Why are many trigonometric integrals well suited for the method of substitution? Integral Calculus Chapter : Integration methods Section 7: Trigonometric integrals by basic methods Page

6 What questions do you have for your instructor? Integral Calculus Chapter : Integration methods Section 7: Trigonometric integrals by basic methods Page 6