Jan. 12 Trig Substitution

# Jan. 12 Trig Substitution - The function sec θ is 1-1 on[0...

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Wednesday January 12 Lecture 4 : More Integration Methods : Trigonometric Substitution ( Refers to Section 7.3 in your text ) Expectations: 1. Solve integrals of functions containing a 2 x 2 , a 2 + x 2 or x 2 a 2 by applying an appropriate trig substitution. 2. Solve definite integrals by trig substitution. 4.1 The method called “Trigonometric substitution” This method applies to integrands containing a 2 x 2 , a 2 + x 2 or x 2 a 2 . It is summarized in the following table: Note The restriction x = a sec θ, 0 ≤ θ < π /2, π ≤ θ < 3 π /2 is due to the way θ = arcsec y is defined in your text. That is, θ = arcsec y if and only if sec θ = y and θ ∈ [0, π /2) [ π , 3 π /2). The definition is not universal. But the important point is that inverse of trigonometric functions are defined on intervals where the function is one-to-one

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Unformatted text preview: . The function sec θ is 1-1 on [0, π /2) ∪ [ π , 3 π /2). Here are some prototype examples that help illustrate how to proceed. 4.1.1 Example 1: 4.1.2 Example 2: 4.1.3 Example 3: 4.2 Example − Find Solution: Remark – Note that the restrictions on the domain of the integrands will often not be expressed explicitly unless it is required by the problem in question. It is a good idea to at least check what the restrictions are in case this is relevant in the problem later on. 4.3 Example − Find Solution: 4.4 Example − Find Solution: 4.5 Example − Compute Solution: 4.6 Note − After having verified that they are indeed true, students can use the following formulas without proving them every time....
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Jan. 12 Trig Substitution - The function sec θ is 1-1 on[0...

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