7.3 - 7.3: Trigonometric Substitution (Dated: October 20,...

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7.3: Trigonometric Substitution (Dated: October 20, 2011) To make the trigonometric substitutions x = a sin θ , x = a tan θ , and x = a sec θ , one needs to restrict θ such that x = x ( θ ) is one-to-one. Also, these trigonometric substitutions are effective for different radical expressions in view of the specified trigonometric identities. 1. x = a sin θ is effective for the radical expression p a 2 - x 2 because 1 - sin 2 θ = cos 2 θ . One usually re- stricts θ to the intervals - π 2 < θ π 2 or - π 2 θ < π 2 . 2. x = a tan θ is effective for the radical expression p a 2 + x 2 because 1 + tan 2 θ = sec 2 θ . One usually re- stricts θ to the interval - π 2 < θ < π 2 . 3. x = a sec θ is effective for the radical expression p x 2 - a 2 because sec 2 θ - 1 = tan 2 θ . One usually re- stricts θ to the intervals 0 θ < π 2 or π θ < 3 π 2 . Example 1 [Exercise 7.3.6 in the text] Let x = sec θ , 0 θ < π 2 . Then p x 2 - 1 x = p sec 2 θ - 1 sec θ = p tan 2 θ sec θ = | tan θ | sec θ = tan θ sec θ and dx = sec θ tan θ d θ . It follows that
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This note was uploaded on 11/11/2011 for the course MATH 152.01 taught by Professor Geline during the Fall '09 term at Ohio State.

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7.3 - 7.3: Trigonometric Substitution (Dated: October 20,...

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