Thread: Integration of hyperbolic functions within a very difficult proof

I would be very grateful if someone could help me with my proof. I want to show that


where


Cn is a constant. I begin by squaring the above equation, getting:


Simplifying this equation gives:


Next I integrate. Perhaps an expert in trigonometry might spot other ways of reducing the above equation


It is during the integration of this above equation that I'm having my difficulty. I can integrate the first four terms and the second last term but I'm very unsure about the integrals with hyperbolic functions. Can anyone help me?

I see that my equations have appeared very small, but if you click on them, they will be bigger and easier to read. I hope this doesn't put people off

Ok
you are sure about your simplification right I will tell you how to integrate the terms
the first integral the you can put the constant C out of the integration

sinh^2x = ( e^x - e^-x)^2/4 = ( e^2x - 2 + e^-2x )/4 since sinhx=e^x/2 - e^-x/2

now you can integrate ( e^2x - 2 + e^-2x )/4 = ( e^2x/2 -2x + e^-2x/-2 )/4

the second integral sin^2x = ( 1 - cos2x )/2 since " cos2x=1-sin^2x "
now you can integrate ( 1 - cos2x )/2 = ( x - sin2x/2 )/2

the third integral cosh^2x like the sinhx but
coshx = ( e^x + e^-x )/2
cosh^x = e^2x +2 + e^-2x )/4 I leave it for you to integrate
fourth
cos^2x = ( cos2x + 1)/2 since " cos2x = 2cosx - 1 " I leave it for you

ok " the second last term " you mean cos(ax)sin(ax) this equal sin2(ax)/2 since sin 2x= 2sinx cosx and the integral for sin2(ax)/2 = -cos2x/4

Note : the constant C I put it out of the integral and the constant " an " you should look at it foe example
cosh (an x) = ( e^(an x) + e^-(an x) )/2
cosh^2 (an x) = ( e^2( an x ) + 2 + e^-2 ( an x))/4

I think it clear now ...

this about hyperbolic sin and cos

sinhx coshx = (e^x - e^-x)(e^x + e^-x)/4 = (e^2x - e^-2x )/4 = ( sinh 2x )/2 and the integral for sinh2x = cosh2x/2