I went off on a tangent by trying to use
cos(x + x) = cos(2x) = cos(x)cos(x) - sin(x)sin(x)
sin(x + x) = 2 sin(x) cos(x)
Today, I noticed this integral equals sec(2x)/2 -- from what I recalled of what the students showed me.
It's easy to show the derivative of sec(2x) = 2 sec(2x) tan(2x) by using the chain rule:
I wrote the above with LibreOffice 3 and found it easier to use than MS Word Equation Editor. I then used WinGrab to capture the formulas to an image.
In numerical analysis, integrals are easy to calculate since you are summing areas, so the usefulness of analytic solutions seems unimportant. Derivatives are harder to deal with numerically since you have to calculate a difference and divide. However, I recall reading Russian geophysical papers from the 1960's which showed analytic solutions for some integrals and that is sure more elegant and exact than the numerical solutions (Russia didn't have very many digital computers back then!).
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