I, for the life of me, can't seem to figure this one out. It's from the GRE math subject test.
If we could pull out the from the integral, then it's just product rule and FTC. But that is tied up in there.
Any ideas from the calculus geniuses out there?
Of course [LaTeX ERROR: Convert failed] applies. That's just the fundamental theorem of calculus. What I've been asking all along is what about [LaTeX ERROR: Convert failed] ?
Again, the issue here is the integrand is a function of both [LaTeX ERROR: Convert failed] and [LaTeX ERROR: Convert failed] . Furthermore, these variables aren't easily separable.
The "x" inside the cosine makes this a little more than just the fundamental theorem:
Leibniz's formula extends the fundamental theorem of calculus:
Here, and so that becomes
(Ah, I see that is the same formula that Ackbeet linked to.)
However, it looks to me like the very existence of this function is questionable. What happens to at t= 0, the lower limit of the integral?