∫x*e^x dx/√(e^x -1)不定积分怎么求

令ψ = √(e^x - 1),2ψ dψ = (e^x) dx
= 2∫ ln(ψ² + 1) dψ
= 2ψln(ψ² + 1) - 2∫ ψ dln(ψ² + 1)
= 2ψln(ψ² + 1) - 2∫ ψ • 2ψ/(ψ² + 1) dψ
= 2ψln(ψ² + 1) - 4∫ (ψ² + 1 - 1)/(ψ² + 1) dψ
= 2ψln(ψ² + 1) - 4∫ dψ + 4∫ dψ/(ψ² + 1)
= 2ψln(ψ² + 1) - 4ψ + 4arctan(ψ) + C
= 2x√(e^x - 1) - 4√(e^x - 1) + 4arctan√(e^x - 1) + C
= 2(x - 2)√(e^x - 1) + 4arctan√(e^x - 1) + C