作业帮请高手帮我做2道积分题,
来源：学生作业帮 编辑：作业帮 分类：数学作业 时间：2024/05/06 18:02:10
请高手帮我做2道积分题,
用换元法解 ∫（lnx）^2/x dx
∫tan^3*x dx
用换元法解 ∫（lnx）^2/x dx
∫tan^3*x dx
∫（lnx）^2/x dx=∫（lnx）^2dlnx令t=lnx
则上式=∫t^2dt=(t^3)/3+C=[(lnx)^3]/3+C
∫tan^3*x dx 令t=tanx,则x=arctant,dx=[1/(1+t^2)]dt
原式=∫[t^3/(1+t^2)]dt
=∫[(t^3+t-t)/(1+t^2)]dt
=∫tdt-∫t/(1+t^2)dt
=(t^2)/2-(1/2)∫1/(1+t^2)d(t^2+1)
=(t^2)/2-(1/2)ln(1+t^2)
=tan^2x/2-(1/2)ln(1+tan^2x)
则上式=∫t^2dt=(t^3)/3+C=[(lnx)^3]/3+C
∫tan^3*x dx 令t=tanx,则x=arctant,dx=[1/(1+t^2)]dt
原式=∫[t^3/(1+t^2)]dt
=∫[(t^3+t-t)/(1+t^2)]dt
=∫tdt-∫t/(1+t^2)dt
=(t^2)/2-(1/2)∫1/(1+t^2)d(t^2+1)
=(t^2)/2-(1/2)ln(1+t^2)
=tan^2x/2-(1/2)ln(1+tan^2x)