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Favorite Old Threads - Best Math Thread # 1

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Indicium Physicus
Staff member
Jan 26, 2012
Originally posted by Srengam on 9/5/2011: integrate
My solution:

One way to integrate involves solving a first-order linear ordinary differential equation. First, note that
That, of course, is just the quotient rule for derivatives. You can integrate it once to obtain
Now, if you could get the integrand to look like the integrand I just mentioned, you'd be done. Let's say you write
$$\int\frac{e^{-x/2-\cos(x)}(1-2\cos(x))(3+2\cos(x))}{e^{-x-2\cos(x)}}\,dx.\quad (1)$$
All I've done is write the exponential in the denominator, and then multiplied top and bottom by the new denominator, because I want to get a $v^{2}$ in the denominator. So now I want $v=e^{-x/2-\cos(x)}.$ This forces my quotient rule to look like this:
Equating the numerator of this RHS with the previous numerator of (1) yields the first-order linear ordinary differential equation
The solution to this DE is
Hence, the integration result is
as WolframAlpha yields.