∫u(x,y,z)dr=C∫y2dx+xdy+zdz
y=ex⇒dy=exdx
z=ex⇒dz=exdx
Then
C∫y2dx+xdy+zdz=0∫1e2xdx+xexdx+ex⋅exdx=0∫12e2xdx+0∫1xexdx=e2x∣∣01+0∫1xexdx=e2−e0+0∫1xexdx
Since
0∫1xexdx=∣∣u=xdu=dxdv=exdxv=ex∣∣=xex∣01−0∫1exdx=xex∣01−ex∣01=e−0−e+e0=1
Then
C∫y2dx+xdy+zdz=e2−e0+0∫1xexdx=e2−1+1=e2
Answer: e2
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