Question

有没有大神帮我看一下这个积分怎么解呢..

Answer 1

图中画红线的地方的确是错误的。
dE=ωlμ0I/[2π(r0+lcosθ)]dl
E=∫(0,L)dE
=∫(0,L) ωlμ0I/[2π(r0+lcosθ)]dl
=ωμ0I/(2π)∫(0,L) l/(r0+lcosθ)dl
=ωμ0I/(2π)·(-1/sinθ)∫(0,L) l/(r0+lcosθ)d(lcosθ)
=-ωμ0I/(2πsinθ)·1/cosθ∫(0,L) lcosθ/(r0+lcosθ)d(r0+lcosθ)
=ωμ0I/(2πsinθcosθ)∫(L,0) [(r0+lcosθ)-r0]/(r0+lcosθ)d(r0+lcosθ)
=ωμ0I/(πsin2θ)∫(L,0) [1-r0/(r0+lcosθ)]d(r0+lcosθ)
=ωμ0I/(πsin2θ)·(r0+lcosθ)|(L,0)-ωμ0I/(πsin2θ)·r0ln(r0+lcosθ)|(L,0)
=ωμ0I/(πsin2θ)[(r0+0cosθ)-(r0+Lcosθ)]-ωμ0Ir0/(πsin2θ)[ln(r0+0cosθ)-ln(r0+Lcosθ)]
=ωμ0I/(πsin2θ)·(-Lcosθ)-ωμ0Ir0/(πsin2θ)ln[r0/(r0+Lcosθ)]
=-ωμ0IL/(2πsinθ)-ωμ0Ir0/(πsin2θ)ln[r0/(r0+Lcosθ)]