I'm trying to calculate the improper Fresnel integral ∞∫0sin(x2)dx calculation.
It uses several substitutions. There's one substitution that is not clear for me.
I could not understand how to get the right side from the left one. What subtitution is done here?
∞∫0v21+v4dv=12∞∫01+u21+u4du.
Fresnel integral calculation:
In the beginning put x2=t and then: ∞∫0sin(x2)dx=12∞∫0sint√tdt
Then changing variable in Euler-Poisson integral we have: 2√π∫∞0e−tu2du=1√t
The next step is to put this integral instead of 1√t.
∞∫0sin(x2)dx=1√π∞∫0sin(t)∫∞0 e−tu2dudt=1√π∞∫0∞∫0sin(t)e−tu2dtdu
And the inner integral ∞∫0sin(t)e−tu2dt is equal to 11+u4.
The next calculation: ∞∫0du1+u4=∞∫0v2dv1+v4=12∞∫01+u21+u4du=12∞∫0d(u−1u)u2+1u2
=12∞∫−∞ds2+s2=1√2arctans√2|∞−∞=π2√2
In this calculation the Dirichle's test is needed to check the integral ∫∞0sint√tdt convergence. It's needed also to substantiate the reversing the order of integration (dudt=dtdu). All these integrals exist in a Lebesgue sense, and Tonelli theorem justifies reversing the order of integration.
The final result is 1√ππ2√2=12√π2
Answer
Well, if one puts v=1u then:
I=∫∞0v21+v4dv=∫∞011+u4du
So summing up the two integrals from above gives:
2I=∫∞01+u21+u4du
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