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Integral Involving Trigonometric Functions with Varying Arguments 
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#1
Mar1410, 11:51 PM

P: 17

1. The problem statement, all variables and given/known data
I'm in an Intermediate Mechanics course right now, and while the Physics itself isn't giving me too much trouble, I am lagging behind in the Math department. I am trying to solve the integral: [tex]\int cos(\omega t) sin(\omega t  \delta) dt[/tex] 2. Relevant equations [tex]sin(AB) = sin(A)cos(B)  sin(B)cos(A)[/tex] 3. The attempt at a solution The first thing I recognized is that the trig functions had the same argument, plus a value, so I figured I could apply the above equation to the integral. However, that really just made things look more complicated. [tex]\int cos(\omega t)sin(\omega t)cos(\delta) sin(\delta)cos^2(\omega t) dt[/tex] I stared at this for a while, but I couldn't find any substitutions (Which is what I was expecting.) I then thought that maybe I should try an integral table, to see if this was listed somewhere, but I couldn't find any functions that might have made sense. The added value in the argument of the "Sin" function is what's tripping me up. Can anyone give me a push in the right direction? 


#2
Mar1510, 12:03 AM

HW Helper
P: 2,322

sin(wt)cos(wt) is easy enough to integrate; just use u=sin(wt). For cos^2(wt), the standard way of integrating this is to use the identity cos(2x)=2cos^2(x)  1.



#3
Mar1510, 12:07 AM

HW Helper
P: 5,003

[tex]\int \left[\cos(\omega t)\sin(\omega t)\cos(\delta) \sin(\delta)\cos^2(\omega t)\right]dt= \cos(\delta)\int\sin(\omega t)\cos(\omega t)dt\sin\delta\int\cos^2(\omega t)dt[/tex] The first integral can be easily done by substituting [itex]u=\sin(\omega t)[/itex] the second integral can be evaluated by using another trigonometric identity, [itex]\cos^2(x)=\frac{1}{2}\left(\cos(2x)+1\right)[/itex] 


#4
Mar1510, 12:09 AM

P: 17

Integral Involving Trigonometric Functions with Varying Arguments
So I am able to rewrite
[tex] \int cos(\omega t)sin(\omega t)cos(\\delta) sin(\delta)cos^2(\omega t) dt [/tex] as [tex] \int cos(\omega t)sin(\omega t)cos(\\delta)dt  \int sin(\delta)cos^2(\omega t) dt [/tex] ? I guess that does make it really easy  thanks =) 


#5
Mar1510, 12:10 AM

P: 17

Sorry for the redundant information  I posted at nearly the same time as you did.
Thanks very much for your help =). 


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