Quantum Mechanics Integral Trouble

jakePHYS
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1. Problem: int[(x^2)*(sin^2 (n*Pi*x/a)] from 0 to a

3. Physics teacher tells me to just look it up. I've wrestled with every table of integrals I can find (web, library, my books), but I'm still hung up.
 
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Write it as x^2*sin(k*x) where k is n*pi/a. First use the trig identity sin^2(kx)=(1-cos(2kx))/2. Then to integrate something like x^2*cos(2kx) you just have to integrate by parts twice. It's not the easiest integral in the world, but it's not the hardest either.
 
Thnx

:biggrin: This has been vexing me for 3 1/2 weeks now! THANK YOU! :biggrin:
 

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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