Can the Intermediate Value Theorem Prove the Infinite Roots of tan x - x = 0?

  • Thread starter Thread starter transgalactic
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You have to show us some work!

Basically what I understand is that you want to show that tan x -x = 0 has an infinite number of roots using the Intermediate Value Theorem. Basically, what they are doing is taking advantage of the pi-periodicity of the tangent function. To roughly understand it - within every period, x only goes through a range of pi, while the tan function goes through a range from -infinity to +infinity. Hence, it is easy to find values of x for which tan x - x is positive, and easy to find it when its negative as well.

Try putting those ideas into equations.
 

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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