Question

∫(0,1) [x^3 * (1-x^2)^0.5 ]dx

Answer 1

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.