Integral Involving Trigonometric Functions with Varying Arguments

In summary, the student is trying to solve an integral, but is having trouble with the trigonometric functions. He is able to solve the first part by using substitution, but is having trouble with the second part. He is able to solve the second part by using another trigonometric identity.
  • #1
Joshk80k
17
0

Homework Statement



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]


Homework Equations



[tex]sin(A-B) = sin(A)cos(B) - sin(B)cos(A)[/tex]


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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  • #2
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
Joshk80k said:
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]
 
  • #4
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
Sorry for the redundant information - I posted at nearly the same time as you did.

Thanks very much for your help =).
 
  • #6
Joshk80k said:
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].
 

1. What is an integral involving trigonometric functions with varying arguments?

An integral involving trigonometric functions with varying arguments is a mathematical expression that involves the integration of a function that contains both a trigonometric function (such as sine or cosine) and a varying argument (such as x or t).

2. How do you solve an integral involving trigonometric functions with varying arguments?

The solution to an integral involving trigonometric functions with varying arguments depends on the specific function and argument involved. In general, techniques such as substitution, integration by parts, and trigonometric identities can be used to simplify the integral and find a solution.

3. What are some common examples of integrals involving trigonometric functions with varying arguments?

Some common examples of integrals involving trigonometric functions with varying arguments include ∫ sin(x)dx, ∫ cos(ax)dx, and ∫ tan(x)dx. These integrals can also have different variations, such as adding a constant term or including other trigonometric functions.

4. What is the significance of integrals involving trigonometric functions with varying arguments?

Integrals involving trigonometric functions with varying arguments have many applications in physics, engineering, and other areas of science. They are often used to model and solve real-world problems involving periodic functions, such as the motion of a pendulum or the behavior of waves.

5. Are there any special techniques for solving integrals involving trigonometric functions with varying arguments?

Yes, there are several special techniques that can be used to solve specific types of integrals involving trigonometric functions with varying arguments. These include the use of trigonometric identities, u-substitution, and integration by parts. It is important to carefully analyze the integral to determine the best approach for solving it.

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