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8.5.7-eg-1

8.5.7-eg-1 - double angle formula to compute Z √ 9-x 2 dx...

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Example Evaluate Z q 9 - x 2 dx . 1. Use a trigonometric substitution The expression under the radical can written as 9 - x 2 = 9 1 - x 3 2 ! = 9(1 - sin 2 t ) = 9(cos 2 t ) , if we set x 3 = sin t which leads to the trig substitution x = 3 sin t, dx = 3 cos t dt. Since the integrand is well-defined for all x such that 9 - x 2 0, or solving the quadratic inequality, for all - 3 x 3, the substitution holds when - 1 sin t 1, or for - π 2 t π 2 . We rewrite the integral as Z 9 - x 2 dx = Z 9 q 1 - sin 2 t cos t dt = 9 Z cos 2 t cos t dt = 9 Z cos 2 t dt, where we have used that cos 2 t = | cos t | = cos t 0 for all - π 2
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Unformatted text preview: double angle formula to compute Z √ 9-x 2 dx = 9 Z cos 2 t dt = 9 Z 1 + cos 2 t 2 dt = 9 t 2 + 9 sin 2 t 4 + C. 2. Back substitute to the original variable. Using the double angle formula sin 2 t = 2 sin t cos t and the substitution x = 3 sin t , it follows that t = arcsin ³ x 3 ´ , sin t = x 3 and cos t = cos ³ arcsin ³ x 3 ´´ = √ 9-x 2 3 (drawing a right triangle). Thus, Z √ 9-x 2 dx = 9 2 arcsin ± x 3 ² + x √ 9-x 2 2 + C....
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