Question

数学积分问题

数学积分问题求解详细步骤和背后知识点。

Answer 1

1) ∫ v*lnvdv
=(1/2)∫lnvdv²
=(1/2) [ v²lnv - ∫v²dlnv]
=(1/2) [ v²lnv - ∫vdv]
=(1/2) [ v²lnv - (1/2)v²]
=(1/2)v²lnv - (1/4)v² + c

2) ∫(v+1)/(1-v)dv
= - ∫[ 1 + 2/(v-1)]dv
= - ∫dv - ∫2/(v-1)d（v-1）
= -v - 2ln(v-1) + c

3) ∫(1-v)/(1+v²)dv
= ∫1/(1+v²)dv - ∫v/(1+v²)dv
= arctanv - (1/2)∫1/(1+v²)dv²
= arctanv - (1/2)∫1/(1+v²)d（1+v²）
= arctanv - (1/2)ln（1+v²） + c