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8.4.15-eg-1 - Example Evaluate 1 cos x dx 0 Two solutions...

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Example Evaluate: Z π 0 1 - cos x dx . Two solutions are shown below: 1. First solution: Using the double angle formula, sin 2 θ = 1 - cos 2 θ 2 , or in this case, 2 sin 2 x 2 = 1 - cos x. Thus, Z π 0 1 - cos x dx = Z π 0 s 2 sin 2 x 2 dx = 2 Z π 0 sin x 2 dx, using the fact that q y 2 = | y | . Then since sin x 2 0 for x [0 , π ], Z π 0 1 - cos x dx = 2 Z π 0 sin x 2 dx = - 2 2 cos x 2 π 0 = - 2 2 (0 - 1) = 2 2 . 2. Second solution: Z π 0 1 - cos x dx = Z π 0 1 - cos x 1 · 1 + cos x 1 + cos x dx = Z π 0 1 - cos 2 x 1 + cos x dx = Z π
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