Integrate ∫ cos(1/x)/x^2 dx: Steps & Solutions

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∫ (cos (1/x) / x^2 dx (Please show steps.)
 
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You show us *your* steps, and then we can offer tutorial help. We do not do your work for you here on the PF.
 
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\int \frac{cos \frac{1}{x}}{x^2} dx
Why couldn't you try the method of substitution? Find an appropriate substitution, and then you need to find something that can serve as du or rather, something that can simplify the denominator.

I hope you enjoy yourself!
 

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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