I have a problem with the following integral

avalg10o

avalg10o

Answered question

2022-03-30

I have a problem with the following integral
xsinxx4+1

Answer & Explanation

crazyrocketrz5z

crazyrocketrz5z

Beginner2022-03-31Added 11 answers

We first write the integral of interest as
xsinxx4+1dx=Im(xeixx4+1dx) (1)
Next, we analyze the contour integral
Czeizz4+1dz=RRxeixx4+1dx+0πReiϕeiReiϕR4ei4ϕ+1iReiϕdϕ (2)
As R, the second integral in (2) vanishes while the imaginary part of the first integral becomes the integral on the right-hand side (1). Therefore, we have
xsinxx4+1dx
=Im(2πiRes(zeizz4+1,z=eiπ4,ei3π4))
The residues at z=±1+i2 are given by
The sum of the residues is therefore
12e12sin(12)
Putting it all together yields
xsinxx4+1dx=πe12sin(12)

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