∫ln[x+(1+x^2)^(1/2)]dx(分部积分法怎么求)

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∫ln[x+(1+x^2)^(1/2)]dx(分部积分法怎么求)
∫ln[x+(1+x^2)^(1/2)]dx(分部积分法怎么求)

∫ln[x+(1+x^2)^(1/2)]dx(分部积分法怎么求)
∫ln[x+(1+x^2)^(1/2)]dx
=xln[x+(1+x^2)^(1/2)]-∫[x(1+x/(1+x^2)^(1/2)]/[x+(1+x^2)^(1/2)]dx
=xln[x+(1+x^2)^(1/2)]-∫[x/(1+x^2)^(1/2)]dx
=xln[x+(1+x^2)^(1/2)]-1/2∫(1+x^2)^(-1/2)d(1+x^2)
=xln[x+(1+x^2)^(1/2)]-(1+x^2)^(1/2)+C
=xln[x+√(1+x^2)]-√(1+x^2)+C

∫ln[x+(1+x^2)^(1/2)]dx
=xln[x+(1+x^2)^(1/2)]-∫xdln[x+(1+x^2)^(1/2)]
=xln[x+(1+x^2)^(1/2)]-∫x/[x+(1+x^2)^(1/2)]*[1+x/(1+x^2)^(1/2)]dx
=xln[x+(1+x^2)^(1/2)]-∫x/(1+x^2)^(1/2)dx
=xln[x+(1+x^2)^(1/2)]-(1+x^2)^(1/2)+C