integral sin(1/x) from 0 to 1
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Integral in addition to its meaning as "antiderivative" is the area beneath a curve. The curve sin(1/x) is not undefined when x=0 because sin(infinity) is undefinable, even though the sine function ranges between -1 and 1. When x=2/(pi), sin(1/x)=1; when x=1/(pi) sin(1/x)=0. At x=1/2(pi) the function is zero again and goes to zero for every x=1/n(pi), where n is an integer. If the curve is undefined in the given range it follows that the area under the curve is not defined either. As x approaches 0, the sine wave "concertinas" with increased frequency, but with amplitude 2 (swinging between 1 and -1) bunching up towards the origin. The area fluctuates between positive and negative with increased intensity and it could be argued that the positive areas above the x axis and the negative areas below the axis cancel each other out, so the net area is zero near the origin.

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