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I've recently discovered the sine integral and have been playing around with it a bit on some graphing software. I looked at the graph of ##Si(x^2) - \frac π 2## and saw that both the amplitude and period was decreasing as x increased. Curiosity got the best of me so I decided to calculate the integral of the function from 0 to infinity to find the area enclosed between the function and the x-axis, which turned out to be ##-\sqrt\frac π 2## square units. This makes sense due to the large portion of the graph being under the x-axis, however I was still curious to find out what the area unde the graph would be if the function was only above the x-axis so I tried integrating the absolute of the function. However here's where I had some trouble; nothing seems able to find the integral of this, not even Wolfram alpha (at least with standard computation time).

So long story short...

##\int_0^∞ Si(x^2) - \frac π 2 dx## = ##-\sqrt \frac π 2##

I want to find out what ##\int_0^∞ |Si(x^2) - \frac π 2 |dx## is equal to. I have tried using heaps of online integral calculators but none of them seem to be able to do it. Wolfram Alpha times out but I'm only a standard member, maybe someone with Pro can give it a go. My intuition tells me that the answer is going to be something like ##\sqrt π## but I'm really not sure if this is right or not.

Any help with this is greatly appreciated :)