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0:56
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An Oscillating Integral: Differentiate, Substitute, Count the Periods
Evaluate the integral 1 I=β« |d cos(ln (1/x))/dx| dx, ,nβZ (positive integer). e^(β2nΟ) Since ln(1/x)=βlnx and cosine is even, differentiating gives d cos(lnx)/dx=-sin(lnx)/x. Using u=βlnx, the integral becomes 2nΟ β« β£sinuβ£du. 0 Because β£sinuβ£ has period Ο, with area 2 over each period, the interval contains 2n complete ...
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