I have ## \int_{t = 0}^{t = 1} \frac{1}{x} \frac{dx}{dt} dt = \int_{t = 0}^{t = 1} (1-y) dt ## [1](adsbygoogle = window.adsbygoogle || []).push({});

The LHS evaluates to ## ln \frac{(x(t_0+T))}{x(t_0)} ##, where ##t_{1}=t_{0}+T##

My issue is that, asked to write out the intermediatary step, I could not. I am unsure how you do this when the limits aren't expressed in terms of ##x## here. So I can see that ##1/x dx = ln x ##, but I'm unsure of what has happened to ## '\frac {dt}{dt}' ## and how the integration limits are done properly.

I think the two below points tie in, and link to where my understanding is lacking with this:

1) Am I correct in thinking that, given ##dx/dt = x(1-y) ##, in order to get to expression [1] you can not simply, formally, 'seperate variables' - see point below, but rather you should divide by ##x## and then integrate both sides w.r.t ##t##.

2) From high school up to now, I have used seperation of variables to solve such things as : ##dx/dt=b(x)## => ##dx/b(x)=dt##

However my lecturer today told me that is inproper and rather one should use a integration factor - i think understanding this point ties with/ may help my initial question?

Many thanks in advance.

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# I Integration - chain rule / functional

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