The integral ∞∫0dx√1+x4 is equal to Γ(14)24√π.
It is calculated or verified with a computer algebra system that Γ(14)24√π=K(12) , where K(m) is the complete elliptic integral of the first kind. This is in relation to what is called the elliptic integral singular value.
It is also known or verified that
K(12)=∫π201√1−sin2(t)2dt=12∫π201√sin(t)cos(t)dt.
Can one prove directly or analytically that
∞∫0dx√1+x4=12∫π201√sin(t)cos(t)dt=∫π201√1−sin2(t)2dt=K(12) ?
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