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Unformatted text preview: Trigonometric Substitutions Working with Right Triangles a 2 2 b a + a b a c 2 2 a c When the square root contains a difference, it represents a leg. When the square root contains a sum, it represents the hypotenuse. In this Section We will learn to evaluate integrals that involve either the sum or the difference of squares. We will use the following facts: hypotenuse side opposite sin = A A side adjacent side opposite tan = A side adjacent hypotenuse sec = A Examples dx x ∫ 2 4 . 1 2 4 x θ There is no integration formula for this integral and no usubstitution can be used. The difference of squares under the radical indicates that a right triangle can be used. 2 x If the variable is opposite to an acute angle, we should select that angle to work with. θ θ sin 2 2 sin = ⇒ = x x θ θ sin 2 2 sin = ⇒ = x x Since we will be replacing x by an expression involving a trigonometric function, we are making a trigonometric substitution. θ θ θ θ θ cos 2 4 2 4 cos cos 2 sin 2 2 2 = ⇒ = = ⇒ = x x d dx x dx x ∫ 2 4 ∫ ∫ ∫ = = ⇒ θ θ θ θ θ d d dx x 2 2 cos 4 cos 2 cos 2 4 Once the trigonometric substitution is complete, we obtain a trigonometric integral, as the ones studied in the last section. [ ]...
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This note was uploaded on 01/21/2011 for the course PHYS 4A 60865 taught by Professor L. oldewurtel during the Fall '09 term at Irvine Valley College.
 Fall '09
 L. OLDEWURTEL

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