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Integral Involving Trigonometric Functions with Varying Arguments

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Joshk80k
#1
Mar14-10, 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(A-B) = 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?
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ideasrule
#2
Mar15-10, 12:03 AM
HW Helper
ideasrule's Avatar
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.
gabbagabbahey
#3
Mar15-10, 12:07 AM
HW Helper
gabbagabbahey's Avatar
P: 5,003
Quote Quote by Joshk80k View Post
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?
Looks good so far, now just split the integral into two and pull the constants out front:

[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]

Joshk80k
#4
Mar15-10, 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 =)
Joshk80k
#5
Mar15-10, 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 =).
gabbagabbahey
#6
Mar15-10, 12:11 AM
HW Helper
gabbagabbahey's Avatar
P: 5,003
Quote Quote by Joshk80k View Post
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 =)
Sure, one of the fundamental rules of calculus is that [itex]\int\left[f(x)+g(x)\right]dx=\int f(x)dx+\int g(x)dx[/itex].


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