Improper integral with variable

DIrtyPio
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With the help of F:[0-> infinity) F(t)= S( (e^(-tx)) sin(2x)/x )dx
find the S sin(2x)/x dx . The integral goes from 0 to infinity.
 
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Note that they are basically asking you to calculate F(0).
 
And can you think of a way to write a simple expression for F'(t)? Use that to find F(t).
 
Sorry, indeed I figured out how should I find the other integral, but actually I'm having problems integrateing F'(t).
 
Sorry to hear that. Maybe if you show what kind of problems, someone could help.
 

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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