Intgration of a harmonic function help please

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intgration of a function*** help please***

actually this is not a homework, I found this explanation in a journal paper but I could not understand it. Can someone give me an explanation or possibly a proof that:


Homework Statement



if:
e;x}=\sqrt{2}\sum_{h=1}^{H}h\omega&space;V_{h}cos%28h\omega&space;t+\frac{\pi&space;}{2}%29.png


then why integration over whole period is:
t%29}{\mathrm{d}&space;t}&space;\right&space;%29^{2}dt=\omega&space;\sum_{h=1}^{H}h^{2}V_{h}^{2}.png



Homework Equations



I have problem with the power of omega, my solution returns w with power 2, while the power of omega in answer is one, Can someone help me for the reason?
 
Last edited:
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Well, when you do a change of variable in the integral in the LHS, one omega pops up in the denominator, canceling one of the 2 from the numerator. That's why the RHS contains omega only to the power 1.
 


bigubau said:
Well, when you do a change of variable in the integral in the LHS, one omega pops up in the denominator, canceling one of the 2 from the numerator. That's why the RHS contains omega only to the power 1.

thank you, but as you see cos(hwt) contains both h and w. means by changing the variable the power of w and h should be equal. I'm confused:cry:

png.png


and over whole period:

png.png


then we will have

png.png


not

png.png


am I wrong??
 
Last edited:

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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