作业帮不定积分题
来源：学生作业帮助网 编辑：作业帮 时间：2024/05/04 08:05:28
不定积分题
不定积分题
不定积分题
-1/2∫x^2d(arccosx)^2=-1/2x^2(arccosx)^2+1/2∫(arccosx)^2dx^2=-1/2x^2(arccosx)^2+∫x(arccosx)^2dx=
令arccosx=t，x=cost，dx=-sintdt
∫x(arccosx)^2dx=-∫cost*t^2sintdt=-1/2∫t^2sin2tdt=1/4∫t^2dcos2t=1/4t...
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-1/2∫x^2d(arccosx)^2=-1/2x^2(arccosx)^2+1/2∫(arccosx)^2dx^2=-1/2x^2(arccosx)^2+∫x(arccosx)^2dx=
令arccosx=t，x=cost，dx=-sintdt
∫x(arccosx)^2dx=-∫cost*t^2sintdt=-1/2∫t^2sin2tdt=1/4∫t^2dcos2t=1/4t^2cos2t-1/4∫cos2tdt^2=1/4t^2cos2t-1/2∫tcos2tdt=1/4t^2cos2t-1/4∫tdsin2t=1/4t^2cos2t-1/4tsin2t+1/4∫sin2tdt=1/4t^2cos2t-1/4tsin2t-1/8cos2t=1/4(arccosx)^2(2x^2-1)-1/2x√(1-x^2)arccosx-1/8(2x^2-1)
原式=-1/2x^2(arccosx)^2+1/4(arccosx)^2(2x^2-1)-1/2x√(1-x^2)arccosx-1/8(2x^2-1)+C
=-(1/4arccosx)^2-1/2x√(1-x^2)arccosx-1/8(2x^2-1)+C
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