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Integration using trigonometric substitutions
Published in John Bird, Bird's Engineering Mathematics, 2021
Calculus is the most powerful branch of mathematics. It is capable of computing many quantities accurately which cannot be calculated using any other branch of mathematics. Many integrals are not ‘standard’ ones that we can determine from a list of results. Some need substitutions to rearrange them into a standard form. There are a number of trigonometric substitutions that may be used for certain integrals to change them into a form that can be integrated. These are explained in this chapter which provides another piece of the integral calculus jigsaw.
Integration using trigonometric substitutions
Published in John Bird, Engineering Mathematics, 2017
Calculus is the most powerful branch of mathematics. It is capable of computing many quantities accurately, which cannot be calculated using any other branch of mathematics. Many integrals are not ‘standard’ ones that we can determine from a list of results. Some need substitutions to rearrange them into a standard form. There are a number of trigonometric substitutions that may be used for certain integrals to change them into a form that can be integrated. These are explained in this chapter which provides another piece of the integral calculus jigsaw.
Integrating rational functions of sine and cosine using the rules of Bioche
Published in International Journal of Mathematical Education in Science and Technology, 2022
In cases where the ‘obvious’ trigonometric substitution to be used in not forthcoming, a set of rules known as Bioche's rules can be used as a guide. The rules tell you if one of the trigonometric substitutions , or can be used, and if so, which one should be used. They are named after the French mathematician Charles Bioche (1859–1949) who first developed the rules in 1902 in connection to solving trigonometric equations (Bioche, 1902) (see also Babbitt, 1913 for a further account of these rules to the solution of trigonometric equations). While the rules seem to make an occasional appearance in the French mathematical literature under the name of Bioche (Mercier, 2006, pp. 373–376), given their simplicity, they deserve to be better known. Without any name attached, the rules are listed in Zwillinger (1992, p. 108) while more recently the present author presented them in Stewart (2018, pp. 190–191) but gave no formal justification for the reason behind why they work. An expert integrator probably already has an intuitive feeling for the type of trigonometric substitution that will work when it comes to integrating rational functions consisting of sine and cosine. For novices, including most beginning calculus students, having an additional guiding hand is always helpful.
A support learning programme for first-year mathematics*
Published in International Journal of Mathematical Education in Science and Technology, 2019
Poh Wah Hillock, R. Nazim Khan
A key feature of the SLT discourse is the use of questions to elucidate students’ understanding of concepts. Stein (2007) identifies three levels of questions in mathematical discourse: direct questions, open-ended questions and students asking each other questions. We give an example in the SLT context: Direct question: ‘How would you calculate the following integral: ?’ Many students will attempt to determine an antiderivative of , often using incorrect integration techniques which are then addressed by the tutor. The tutor points out that finding an antiderivative of involves some work using trigonometric substitution. ‘Is there an easier way to evaluate the integral?’Open-ended question: ‘What does mean?’ The tutor prompts students by drawing attention to the integral notation as an elongated ‘S’ (for sum).Student–student question: Students engage with each other by discussing the direct and open-ended questions posed by the tutor. Through questions and peer discussion, the aim is for students to arrive at the area interpretation of the definite integral . The answer for the definite integral is then obvious (1/4 the area of the unit circle centred at the origin).
An alternative way of defining integration in multivariable calculus
Published in International Journal of Mathematical Education in Science and Technology, 2022
This example is an application of (14). Let E be the open region in bounded above by the paraboloid and below by the plane . Choose . We have and . To find and , we ‘project’ both surfaces onto the -plane. This entails finding the curves of intersection of the two surfaces, i.e. equating the two surfaces gives . Hence and . Finally, to find a and b, we note that and determine a Type I region in . The points of intersection of and are and , so that a = 0 and b = 2. Observe that for every . Therefore Note that the last integral can be evaluated by rewriting and making the substitution , and then the trigonometric substitution , where .