Question

请大神解答44题，需要详细的解题过程与解析，越详细越好。谢谢！！！

Answer 1

令x=cost
x：0→√3/2，则t：π/2→π/6
∫[0：√3/2][xarccosx/√(1-x²)]dx
=∫[π/2：π/6][tcost/√(1-cos²t)]d(cost)
=∫[π/2：π/6](-tsintcost/sint)dt
=∫[π/6：π/2]tcostdt
=∫[π/6：π/2]td(sint)
=tsint|[π/6：π/2]-∫[π/6：π/2]sintdt
=(π/2)sin(π/2) -(π/6)sin(π/6)+cost|[π/6：π/2]
=(π/2)·1 -(π/6)·½+cos(π/2)-cos(π/6)
=π/2 -π/6 +0 -√3/2
=(2π-3√3)/6