∭cos(x2+y2+z2)3/2dxdydz
Convert x,y,z to spherical coordinates r,φ,θ.
x=rcos(φ)sin(θ)y=rsin(φ)sin(θ)z=rcos(θ)
The absolute value of Jacobian is r2sin(θ). Then the original integral is
∭cos((r2)3/2)⋅rsin(θ)drdφdθ=∭r2cos(r3)sin(θ)drdφdθ=∫r2cos(r3)dr∫dφ∫sin(θ)dθ=31∫cos(r3)d(r3)∫dφ∫sin(θ)dθ=−31φsin(r3)cos(θ)We have omitted the constants for convenience.
Return to x,y,z.
r=x2+y2+z2φ=arctan(xy)θ=arccos(x2+y2+z2z)
Therefore, the original integral is
∭cos(x2+y2+z2)3/2dxdydz=−31arctan(xy)sin((x2+y2+z2)3/2)cos[arccos(x2+y2+z2z)]=−31arctan(xy)sin((x2+y2+z2)3/2)x2+y2+z2z
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