积分下限0上限1/根号下2 arccosx/ [根号(1-x^2)^3] dx

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积分下限0上限1/根号下2 arccosx/ [根号(1-x^2)^3] dx

积分下限0上限1/根号下2 arccosx/ [根号(1-x^2)^3] dx
积分下限0上限1/根号下2 arccosx/ [根号(1-x^2)^3] dx

积分下限0上限1/根号下2 arccosx/ [根号(1-x^2)^3] dx
令x=cost
则原式=∫(π/2→π/4)t/sin^3(t)*(-sint)dt=∫(π/4→π/2)t/sin^2(t)dt=-∫(π/4→π/2)td(cott)=-tcott|(π/4→π/2)+∫(π/4→π/2)cottdt=0+π/4-∫(π/4→π/2)d(sint)/sint=π/4-ln|sint||(π/4→π/2)=π/4-0+ln(√2/2)=π/4+ln(√2/2)