Solving a Tricky Integration Problem: Physics Forum's First-Timer's Attempt

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Hi Physics Forums,

I'm new here, this is my first post here.

I have found this question in an old textbook, after a few hours of trying to solve it I get to a dead end and I have no idea about where to go from there. I am not sure whether I'm missing something simple, or if this could be a really mean question :) So here goes...

1. ^{\frac{\pi}{2}}_{0}\int3^{sinx}dx

Ive attached my attempt so far, I haven't quite got the idea of the posting buttons.

Thank-you so much for your time, I really appreciate it!

Thanks

Tom
 

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Hello Tom, welcome to PF. There is no way that integral can be worked with usual techniques you would learn in calculus. You could always approximate it numerically with, for example, Simpson's rule.
 
Hi

Thanks for such a prompt reply LCKurtz.

Ok then, maybe I shall go onto a new question :)

Just out of interest if we used techniques beyond calculus, how would it be solved?

Thanks so much for all your help,

Tom
 
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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