A plane curve is parameterized by x(t) = [tex]\int^{\infty}_{t}[/tex] [tex]\frac{cos u}{u}[/tex] du and y(t) = [tex]\int^{\infty}_{t}[/tex] [tex]\frac{sin u}{u}[/tex] for 1 [tex]\leq[/tex] t [tex]\leq[/tex] 2. What is the length of the curve?(adsbygoogle = window.adsbygoogle || []).push({});

I know the formula for the length of the curve so I know you need to find dy/dt and dx/dt and integrate the square root of the sum of their squares and integrate that over the interval [1,2].

But my question is finding each derivative. I understand that you could apply the fundamental theorem of calc that in essences states the derivative of the integral is the original function. So you just flip the limits of integration, put a negative out front, and simply write dx/dt = -cos u / u. But in the actual statement of this FTC, the lower limit of integration is a constant (commonly denoted a or c), so I'm second guessing myself. Should it matter that the lower limit is infinity? I guess I was thrown a bit by the infinity and our AP class just covered the FTC before I attempted this problem, it seems like the infinity wouldn't affect the differentiability of an already integrated function.

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# HMMT Calculus Problem

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