I am stuck at this question. Find a closed form (that may actually contain the Gamma function) of the integral
∫∞0sin(xp)dx
I am interested in a Laplace approach, double integral etc. For some weird reason I cannot get it to work.
I am confident that a closed form may actually exist since for the integral:
∫∞0cosxadx=πcscπ2a2aΓ(1−a)
there exists a closed form with Γ and can be actually be reduced further down till it no contains no Γ. But trying to apply the method of Laplace transform that I have seen for this one , I cannot get it to work for the above integral that I am interested in.
May I have a help?
Answer
If p>1,
I(p)=∫+∞0sin(xp)dx=1p∫+∞0x1p−1sin(x)dx
but since L(sin(x))=1s2+1 and L−1(x1/p−1)=s−1/pΓ(1−1p) we have:
I(p)=1pΓ(1−1p)∫+∞0s−1/p1+s2ds=π2pΓ(1−1p)sec(π2p).
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