∫e^x√e^x-1/e^x+3dx上限ln5下限0

Question

∫e^x√e^x-1/e^x+3dx上限ln5下限0

Answer 1

I = ∫ [e^x√(e^x-1)]/(e^x+3) dx = ∫ √(e^x-1)/(e^x+3) de^x
设 √(e^x-1)=t,则 e^x=(1+t^2),de^x=2tdt
I = ∫ tdt/(4+t^2) = (1/2)∫ d(4+t/^2)(4+t^2)
= (1/2)[ln(4+t^2)] = (3/2)ln2