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8.4.1-eg-1 - Example Evaluate sin7 x dx 2 When the power of...

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Unformatted text preview: Example Evaluate sin7 x dx. 2 When the power of sine or cosine is odd, we write the integrand as a product and use the Pythagorean identity: sin7 x x x x = sin6 sin = sin2 2 2 2 2 Making the substitution u = cos sin7 x 2 dx = −2 = −2 x 2 3 sin x x = 1 − cos2 2 2 x 1 and du = − sin 2 2 3 sin x . 2 dx, (1 − u2 )3 du (1 − 3u2 + 3u4 − u6 ) du 3 = −2 u − u3 + u5 − 5 x + 2 cos3 = −2 cos 2 17 u +C 7 6 2 x x x − cos5 + cos7 + C. 2 5 2 7 2 ...
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