Derivation of Carrier Concentration (ni)

m_keown2000
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I am following a lecture on carrier concentration and I got to the point where the instructor said that for homework, derive the carrier concentration equation ni, which equals:https://www.physicsforums.com/attachments/81076

To derive ni you need to compute the following integral which I am having trouble solving:

Snapshot1.jpg


where:
Snapshot2.jpg


I tried Integration by parts and no luck. Any help would be greatly appreciated.
 

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90make the variable change assume X = (E-[E][/C]/kT)
 

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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