# Integral of cosine function

by daudaudaudau
Tags: cosine, function, integral
 P: 295 Hi. I have been experimenting a little to come up with the following "conjecture" $$\int_0^{2\pi}d\phi f(a+b\cos\phi)\sin\phi=0$$ where a and b are arbitrary constants and f(x) is any function. Is this true? I guess it can be shown by expanding f in a power series of cosines?
 PF Patron HW Helper Sci Advisor Thanks P: 25,481 hi daudaudaudau! doesn't work for f = √, a = b
 PF Patron P: 864 Yes, I'm getting that that this is true. It can be proven using the substitution $$x=\cos{\phi}$$ $$dx=-\sin{\phi}d\phi$$ But this substitution is not 1-to-1: Each x value corresponds to 2 phi values on [0,2pi]. So you need to break the region of integration into [0,pi] and [pi,2pi]. If you look at the graph of the cos function, you will see it is 1-to-1 on these two intervals and goes from 1 to -1 on [0,pi] and from -1 to 1 on [pi,2pi]. So the integral becomes: $$\int_0^{2\pi}d\phi f(a+b\cos\phi)\sin\phi= \int_1^{-1}-f(a+bx)dx + \int_{-1}^{1}-f(a+bx)dx = \int_{-1}^{1}f(a+bx)dx + \int_{-1}^{1}-f(a+bx)dx = 0$$ Tiny-tim: I get this even in the example you gave.
PF Patron
HW Helper
Thanks
P: 25,481

## Integral of cosine function

hmm … i think i've been misled by the ambiguity of the √ function

yes, we can get it directly from the original integral …

if f = g', then it's ∫ [g(a + bcosφ)]' dφ, = [g(a + bcosφ)]φ=0
PF Patron
P: 864
 Quote by tiny-tim hmm … i think i've been misled by the ambiguity of the √ function yes, we can get it directly from the original integral … if f = g', then it's ∫ [g(a + bcosφ)]' dφ, = [g(a + bcosφ)]φ=02π
Happens to everyone
That is a better way of proving it.
P: 295
 Quote by tiny-tim hmm … i think i've been misled by the ambiguity of the √ function yes, we can get it directly from the original integral … if f = g', then it's ∫ [g(a + bcosφ)]' dφ, = [g(a + bcosφ)]φ=02π
What happened to the sine function?
 PF Patron HW Helper Sci Advisor Thanks P: 25,481 chain rule
 P: 295 clever :-)
 Sci Advisor P: 5,415 The integral from -pi to pi is zero, because you are integrating an odd function. And the integral from -pi to pi equals the integral from 0 to 2pi. QED.

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