解:分享一种解法。
①设x=-t,∴原式=100∫(0,1)(t^4)√[(1-t)/(1+t)]dt=100∫(0,1)(t^4)(1-t)dt/√(1-t^2)。
②设t=siny,dt=cosydy,y∈[0,π/2],
∴∫(0,1)(t^4)(1-t)dt/√(1-t^2)=∫(0,π/2)(1-siny)(siny)^4dy=∫(0,π/2)[(siny)^4-(siny)^5]dy,
而(siny)^4=(1/4)(1-cos2y)^2=(1/8)(3-4cos2y+cos4y)、(siny)^5dy=-[1-(cosy)^2]^2d(cosy)=-[1-(cosy)^2]^2d(cosy)=-[1-2(cosy)^2+(cosy)^4]d(cosy),
∴∫(0,1)(t^4)(1-t)dt/√(1-t^2)={(1/8)[3y-2sin2y+(1/4)sin4y]+[cosy-(2/3)(cosy)^3+(1/5)(cosy)^5]}丨(y=0,π/2)=3π/16-8/15,
∴原式=50(3π/8-16/15)=5(45π/8-128)/12。