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PP 8.3 - Trigonometric Substitutions Working with Right...

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Trigonometric Substitutions
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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.
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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
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Examples dx x - 2 4 . 1 2 4 x - θ There is no integration formula for this integral and no u-substitution 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
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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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[ ] θ θ θ θ d d dx x + = = - ) 2 cos( 1 2 cos
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