Page 195 - ADVANCED MATH F.5

Form Five Form Five

Integration by trigonometric substitutions
Teaching steps
1. Guide students through discussion to recognize integrals of
rational functions whose denominators cannot be factorized
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and cannot be expressed in the form x 2 – a . Help students
2
to recognize common useful trigonometric substitutions which
involve tangent, sine, or cosine.

2. Assist students through discussion to find solutions of integrals
of the form:

⌠
(a)  1 dx (b) ⌠ 1 dx (c) ∫ a 2 x dx− 2 ,

2
⌡ x + a 2 ⌡ a 2 x− 2
where a is a constant
3. In groups, formulate an engaging strategy that will assist
students to solve Examples 9.40 to 9.46 in the Student’s Book.
Allow students to share alternative methods of solving integrals
which involve trigonometric substitutions. Engage students to
use mathematical softwares and scientific calculators to verify
the solutions of the given examples. Advise them to share their
solutions through presentations.

4. Direct students to attempt Exercise 9.8 in the Student’s Book.
Advise them to submit their work. Check the correctness
of their answers, and provide them constructive feedback.
Encourage students to explore additional resources, such as
online tutorials, Symbolab, Mathematica or reference books,
to enrich their understanding and for further practice.

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Mathematics for Advanced Secondary Schools

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