Solving Arctanx=x: A Simple Guide to Solving Trigonometric Equations

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Homework Statement


solve
arctanx=x

Homework Equations





The Attempt at a Solution

 
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Have you tried drawing a graph of each of them?
 
There is not going to be any "formula" for the solution, you will have to use some kind of numeric solution. The simplest is Office Shredder's suggestion- accurately graph y= tan x and y= x and measure the x coordinate where the graphs cross.
 
yes I tried to sketch it and I find only x=0

Are my answer is correct?
 

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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