Integral of cos^2((π)x): Is it \frac{1}{3}?

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In summary, the integral of \int^{\frac{1}{2}}_{0}cos^2((\pi)x) is solved by using the Power Reduction Formulae to simplify the integral to \int_{0} ^ {\frac{\pi}{2}} \left( \frac{1 + \cos (2x)}{2} \right) dx and then using a u-substitution to solve it.
  • #1
americanforest
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Quick question: What is the integral of

[tex]\int^{\frac{1}{2}}_{0}cos^2((\pi)x)[/tex]?

Is it

[tex]\frac{1}{3}(cos^3((\pi)x))\frac{1}{\pi}sin((\pi)x)[/tex]

and then plug in or is there something wrong with that?
 
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  • #2
Cos^2(pix) = 1/2 (1 + Cos(2pix))
This is simply integrated.
 
  • #3
Do you know where I can get a proof of that?
 
  • #4
[tex]\cos x=\frac{e^{ix}+e^{-ix}}{2}[/tex]

Square that and you have your result.
 
  • #5
Or square the power series of cosx and rearange but that's more difficult.
 
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  • #6
Or simply use that

[tex] \cos 2x=\cos^{2} x -\sin^{2} x [/tex]

and

[tex] 1=\cos^{2} x +\sin^{2} x [/tex]

Daniel.
 
  • #7
Or basically, we can use the Power Reduction Fomulae:
[tex]\cos ^ 2 x = \frac{1 + \cos (2x)}{2}[/tex]
[tex]\sin ^ 2 x = \frac{1 - \cos (2x)}{2}[/tex]
-------------
Now, back to your problem:
[tex]\int_{0} ^ {\frac{\pi}{2}} \cos ^ 2 (\pi x) dx = \int_{0} ^ {\frac{\pi}{2}} \left( \frac{1 + \cos (2x)}{2} \right) dx[/tex]
Now, all you need to do is to use a u-substitution to solve it.
Can you go from here? :)
 

Related to Integral of cos^2((π)x): Is it \frac{1}{3}?

1. What is the integral of cos^2((π)x)?

The integral of cos^2((π)x) is equal to x/2 + (sin(2πx))/(4π) + C.

2. Why is the integral of cos^2((π)x) equal to x/2 + (sin(2πx))/(4π) + C?

This is because the integral of cos^2((π)x) can be solved using the double angle formula for cosine and the power rule of integration.

3. How do you prove that the integral of cos^2((π)x) is equal to \frac{1}{3}?

To prove that the integral of cos^2((π)x) is equal to \frac{1}{3}, we can use the substitution method and let u = sin(2πx). This will lead to the integral being equal to u^2/8 + C. Then, substituting u back in and simplifying will result in the final answer of x/2 + (sin(2πx))/(4π) + C, which is equal to \frac{1}{3}.

4. Can you use a graph to visualize the integral of cos^2((π)x)?

Yes, a graph can be used to visualize the integral of cos^2((π)x). The resulting graph would be a parabola with a positive slope, starting at the origin and increasing as x increases. The area under the curve, which represents the integral, would be equal to \frac{1}{3}.

5. Are there any real-life applications of the integral of cos^2((π)x)?

Yes, the integral of cos^2((π)x) has applications in calculating the average value of a periodic function over a given interval. It is also used in physics and engineering to determine the work done by a varying force over a specific distance.

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