8.5 - 8.5 1 8.5 Trigonometric Substitution Three Basic...

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Unformatted text preview: 8.5 1 8.5 Trigonometric Substitution Three Basic Substitutions Recall the derivative formulas for the inverse trigonometric functions of sine, secant, tangent. d dx ( sin 1 x ) = 1 1 x 2 , | x | < 1 (1) d dx ( tan 1 x ) = 1 1 + x 2 (2) d dx ( sec 1 x ) = 1 | x | x 2 1 , | x | > 1 (3) Now focus on the quadratic term in each derivative. 8.5 2 For example, in (1), we focus on the 1 x 2 under the radical. 1 1 x 2 (4) Is there a substitution that can help us with this? Lets try x = sin . Thus sin = x 1 (5) Now we can construct the (reference) triangle for . Thus 1 x 2 1 x It follows that 1 1 x 2 = | sec | In practice, we usually want the substitution in (5) to be invertible. That is, x = sin iff = sin 1 x, 2 2 (6) 8.5 3 Example 1. Trig Substitution using x = sin Evaluate (7) integraldisplay 1 1 x 2 dx If (8) 1 < x < 1 we can let x = sin , / 2 / 2 . Then dx = cos d So that integraldisplay dx 1 x 2 = integraldisplay cos d cos 2 = integraldisplay sec d = ln | sec + tan | + C = ln vextendsingle vextendsingle vextendsingle vextendsingle...
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8.5 - 8.5 1 8.5 Trigonometric Substitution Three Basic...

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