Question

∫arcsinx*arccosxdx永不分积分法怎么求

Answer 1

先化简t=arcsin(x) x=sin(t)
arccos(x)=π/2 -t
∫t(π/2 -t)dsin(t)=t(π/2 -t)sin(t) -∫ sint d(t(π/2 -t))
=t(π/2 -t)sin(t) -∫ (π/2-2t)sint dt
=t(π/2 -t)sin(t) +∫ (π/2-2t) dcos(t)
=t(π/2 -t)sin(t) + (π/2-2t) cos(t)-∫cos(t)d (π/2-2t)
=t(π/2 -t)sin(t) + (π/2-2t) cos(t)+∫2cos(t)dt
=t(π/2 -t)sin(t) + (π/2-2t) cos(t)+2sin(t)+C
=arcsin(x)arccos(x)x+(arccos(x)-arcsin(x))√(1-x²)+2arcsin(x)+C

Answer 2

用部分积分？还是分部积分？

追问

不好意思，写错了，用分部积分法！