∫(0,1) [x^3 * (1-x^2)^0.5 ]dx

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∫(0,1)[x^3*(1-x^2)^0.5]dx∫(0,1)[x^3*(1-x^2)^0.5]dx∫(0,1)[x^3*(1-x^2)^0.5]dx设x=sint,dx=costdt,当x=0时,t

∫(0,1) [x^3 * (1-x^2)^0.5 ]dx
∫(0,1) [x^3 * (1-x^2)^0.5 ]dx

∫(0,1) [x^3 * (1-x^2)^0.5 ]dx
设x=sint,dx=costdt,当x=0时,t=0,x=1,t=π/2,
原式=∫[0,π/2](sint)^3(cost)^2dt
=-∫[0,π/2](sint)^2(cos)^2]d(cost)
=∫[0,π/2][1-(cost)^2](cost)^2d(cost)
=-∫[0,π/2](cos)^2d(cost)+∫[0,π/2(cost)^4d(cost)
=[0,π/2]{-[(cost)^3/3-(cost)^5/5]
=-[0-(1/3-1/5)]
=2/15.

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