Calculating the Integral of 1/sinh^2(x/2): Exploring Coth and Principal Value

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In summary, the conversation is about solving the integral of 1/sinh^2(x/2) from -inf to +inf and whether or not it should be considered a principal value integral. The conversation also clarifies that the integral cannot be a function of x once limits are placed on it, and asks if the goal is to prove the anti-derivative.
  • #1
Belgium 12
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Hi dear members,
I have a problem with the following integral:

Integral 1/sinh^2(x/2) from -inf to +inf =-2 coth(x/2)

must I consider it as a principal value integral?

Thank you
 
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  • #2
If you're integrating wrt x, you can't have x in the RHS.
 
  • #3
Belgium 12 said:
Integral 1/sinh^2(x/2) from -inf to +inf =-2 coth(x/2)

Just to be clear, are you trying to evaluate
##\int_{-\infty}^{\infty} \frac{1}{\sinh ^2(x/2)} \, dx##
 
  • #4
once you put limits on the integral (a definite integral, not an indefinite integral), it is no longer a function of x. is it the anti-derivative you want to prove?
 

FAQ: Calculating the Integral of 1/sinh^2(x/2): Exploring Coth and Principal Value

1. What is the integral of 1/sinh^2(x/2)?

The integral of 1/sinh^2(x/2) is equal to -2coth(x/2) + C, where C is the constant of integration.

2. How is the integral of 1/sinh^2(x/2) derived?

The integral of 1/sinh^2(x/2) can be derived using the substitution method, where u = sinh(x/2) and du = 1/2cosh(x/2)dx.

3. Does the integral of 1/sinh^2(x/2) have any real-life applications?

Yes, the integral of 1/sinh^2(x/2) is used in physics and engineering to calculate the magnetic field of a current-carrying wire.

4. Can the integral of 1/sinh^2(x/2) be solved using other methods besides substitution?

Yes, the integral can also be solved using the trigonometric substitution method, where u = tan(x/2) and du = 1/2sec^2(x/2)dx.

5. Are there any special properties or identities associated with the integral of 1/sinh^2(x/2)?

Yes, the integral of 1/sinh^2(x/2) can be rewritten as -2sech(x/2) + C, which is similar to the integral of 1/cosh^2(x/2). Additionally, it can also be expressed in terms of hyperbolic functions as -2csch(x) + C.

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