Question

已知隐函数怎么求其原函数?如x=y(x-y)^2 怎么求y的不定积分?

Answer 1

令t=x-y y=x-t
x=y(x-y)^2
x=(x-t)t^2
x=xt^2-t^3
x=t^3/(t^2-1)
∫t^3/(t^2-1)dt
=1/2∫(2t+1/(t+1)+1/(t-1)dt
=1/2(t^2+ln(t+1)+ln(t-1)+C
将t=x-y 代入
x=1/2((x-y)^2+ln(x-y+1)+ln(x-y-1)+C