College Mathematics Journal

Vol48-N4

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CLASSROOM CAPSULES EDITORS Ricardo Alfaro Lixing Han Kenneth Schilling University of Michigan–Flint University of Michigan–Flint University of Michigan–Flint ralfaro@umflint.edu lxhan@umflint.edu ksch@umflint.edu Classroom Capsules are short (1–3 page) notes that contain new mathematical insights on a topic from undergraduate mathematics, preferably something that can be directly introduced into a college classroom as an effective teaching strategy or tool. Classroom Capsules should be prepared according to the guidelines on the inside front cover and submitted through Editorial Manager. The Demise of Trig Substitutions? David Betounes (betounes d@utpb.edu), Mylan Redfern (redfern m@utpb.edu), Uni- versity of Texas of the Permian Basin, Odessa, TX Having taught Calculus II for more years than we would like to count, we recently were stunned to discover that there is an alternative to trigonometric substitutions—an alternative that has apparently gone unnoticed by calculus book authors for over three decades (and perhaps longer, but that is the oldest calculus book we could locate). Our shock about the possibility for such an alternative method occurred while grading a recent test. One student's work for R x 3 √ 4 − x 2 d x showed that letting u = 4 − x 2 leads to a solution somewhat more quickly than the customary trigono- metric substitution x = 2 sin θ . But of course! (An occasional calculus book will do this too, but not in the "trig subs" section). Clearly, we thought, this will work for all other odd powers of x . After recovering from this surprise, we began a search for a simple, alternative method for the even power case. This we had never seen done anywhere. We eventually found that the key for doing the even power case lay in redoing the odd power case. Namely, use u 2 = 4 − x 2 for odd powers and u 2 = (4 − x 2 )/x 2 for the even powers. u-square substitutions. There are four types of integrals that we can effectively compute using trigonometric substitutions. They are (a) Z x n R k d x, (b) Z x n R k d x, (c) Z R k x n d x, (d) Z 1 x n R k d x, (1) where n ≥ 0, k > 0 are integers, k is odd, and R = (a 2 − x 2 ) 1/2 , (a 2 + x 2 ) 1/2 , or (x 2 − a 2 ) 1/2 . (2) In all cases, these integrals can be computed (sometimes more quickly) using what, for lack of better name, we call u 2 -substitutions: u 2 = R 2 for n odd, u 2 = R 2 x 2 for n even . (3) http://dx.doi.org/10.4169/college.math.j.48.4.284 MSC: 26A03 284 © THE MATHEMATICAL ASSOCIATION OF AMERICA

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