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Calculus II Notes 8.3

# Calculus II Notes 8.3 - Calculus II-Stewart Dr Berg Spring...

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Calculus II- Stewart Dr. Berg Spring 2010 Page 1 0.0 8.3 Trigonometric Substitution We use trigonometric identities to algebraically simplify functions to be integrated. For example, x x 2 dx can be easily integrated using the method of u substitution, but the area of the upper half of a unit circle x 2 1 1 dx requires a change of variables that allows the use of a Pythagorean identity. Let x = sin θ . Then dx = cos d , and 1 = x = sin implies that = π /2 , and 1 = x = sin implies = . Thus x 2 1 1 dx = sin 2 / 2 / 2 cos ( ) d = cos 2 / 2 / 2 cos ( ) d = cos cos / 2 / 2 d = cos 2 0 / 2 d cos 2 / 2 0 d = 2 cos 2 0 / 2 d = 2 1 2 1 + cos2 x ( ) 0 / 2 d = + sin2 x 2 0 / 2 = 2 . Table of Trigonometric Substitutions Expression Substitution Identity 2 x 2 x = a sin for < < 1 sin 2 = cos 2 2 + x 2 x = a tan for < < 1 + tan 2 = sec 2 x 2 a 2 x = a sec for 0 < < or < < 3 sec 2 1 = tan 2 Example A Evaluate 1 x 2 x 2 + 4 dx .

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Calculus II Notes 8.3 - Calculus II-Stewart Dr Berg Spring...

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