令x=2sint,t∈[-π/2,π/2] 则
√(4-x²)=√(4-4sin²t)=2cost,dx=2costdt
∫1/[x√(4-x²)] dx
= ∫2cost/(4sintcost) dt
=(1/2)∫csctdt
=(1/2)ln|csct-cott|+C
=(1/2)ln|[2-√(4-x²)]/x|+C
C为任意常数
令x=2sint,t∈[-π/2,π/2] 则
√(4-x²)=√(4-4sin²t)=2cost,dx=2costdt
∫1/[x√(4-x²)] dx
= ∫2cost/(4sintcost) dt
=(1/2)∫csctdt
=(1/2)ln|csct-cott|+C
=(1/2)ln|[2-√(4-x²)]/x|+C
C为任意常数