Exploring the Cot Integral: Strategies for Evaluating and Simplifying

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In summary, the question is about evaluating the integral of cot^2(2x) from \pi/4 to \pi/8. There is confusion about the correct limits and whether to use the identity cot(2x)=cos(2x)/sin(2x) or cotx=1/tanx. It is suggested to use the identity and reduce the even powers using trig identities. However, the correct integral is found by using the identity cos^2(2x)+sin^2(2x)=1 and splitting the integral into two parts.
  • #1
Briggs
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I have the question Evaluate [tex]\int_{Pi/4}^{Pi/8}_cot^2{2x}dx[/tex]
So integrating this should (I hope) give [tex][\frac{-1}{2}cot{2x}-x][/tex] for those limits.
But I have never evaluated the cot integral before, I know that cotx=1/tanx. Do I substitute this identity in and work from there?
 
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  • #2
You could but I suspect that [itex]\frac{1}{tan^2 x}[/itex] would not be easy to integrate. How about just [itex]cot(2x)= \frac{cos 2x}{sin 2x}[/itex]?

[tex]\int_{\pi/4}^{\pi/8}cot^2(2x)dx= \int_{\pi/4}^{\pi/8}\frac{cos^2(2x)}{sin^2(2x)}dx[/tex].

Since those are even powers you will need to use trig indentities to reduce them. By the way is there a reason for integrating from a larger value of x to a smaller?
 
  • #3
Ah that was a mistake, the x values should be the other way around.

So would [tex]\int_{\pi/8}^{\pi/4}cot^2(2x)dx= \int_{\pi/8}^{\pi/4}\frac{cos^2(2x)}{sin^2(2x)}dx[/tex] integrate to [tex]\frac{\frac{1}{2}sin^2(2x)}{\frac{-1}{2}cos^2(2x)}[/tex] ? Which I assume would simplify to [tex]-tan^2_(2x)[/tex]
 
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  • #4
Unfortunately not, but use the fact that [itex]\cos ^2 \alpha + \sin ^2 \alpha = 1[/itex] on:

[tex]\int {\frac{{\cos ^2 \left( {2x} \right)}}{{\sin ^2 \left( {2x} \right)}}dx} = \int {\frac{{1 - \sin ^2 \left( {2x} \right)}}{{\sin ^2 \left( {2x} \right)}}dx} [/tex]

Then split the integral in two.
 

1. What is a cot integral?

A cot integral is an integration of the cotangent function, which is the reciprocal of the tangent function. It is used in calculus to find the area under a curve that is defined by the cotangent function.

2. Why is evaluating a cot integral important?

Evaluating a cot integral is important because it allows us to find the value of the area under a curve that is defined by the cotangent function. This is useful in many applications, such as in physics and engineering.

3. How do you evaluate a cot integral?

To evaluate a cot integral, we use integration techniques such as substitution, integration by parts, or partial fractions. It is important to identify the appropriate technique to use based on the form of the integral.

4. What are some common strategies for evaluating a cot integral?

Some common strategies for evaluating a cot integral include using trigonometric identities, converting the cotangent function to a sine or cosine function, and using integration techniques such as u-substitution or integration by parts.

5. Are there any special cases when evaluating a cot integral?

Yes, there are special cases when evaluating a cot integral. One example is when the integral involves a cotangent function raised to an odd power, in which case we can use the power-reducing formula to simplify the integral. Another special case is when the integral involves a cotangent function multiplied by a tangent function, in which case we can use the tangent half-angle formula to simplify the integral.

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