Question

积分计算§X^2（1-X^2）^1/2 §号表示积分 即X的2次方乘以1减X的2次方的

Answer 1

设x=sint,则dx=costdt,t=arcsinx,cost=√(1-x²)

代入原式得,原式=∫sin²tcos²tdt

=1/4∫sin²(2t)dt

=1/8∫(1-cos(4t))dt

=1/8(t-sin(4t)/4)+C (C是积分常数)

=1/8(t-sintcost(cos²t-sin²t))+C

=1/8(arcsinx-x(1-2x²)√(1-x²))+C.