Integral from 0 to 1 of 1/sqrt(1+x^2)

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Homework Statement


\int_0^1 \frac{1}{\sqrt{x^2+1}}\,dx

Homework Equations


Integration by substitution looks like it might help here...

The Attempt at a Solution


The answer is \log (1+\sqrt 2), but I'm at a loss as to how to derive that.
 
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You could try a trig substitution such as x=tan\theta.Hyperbolic trig sub. would be possible as well.
 
Thanks. I worked it out using your suggestion, but x = sinh(theta) also works, in case anyone cares. :-)
 

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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