Question

∫x^3/(x^8-2) dx∫(x^3-1)/(x^2+1) dx

Answer 1

1.令t=x^4 dt4=x^3dx

原式=∫dt4(t^2-2)=2^(12)16∫{dt[t-2^(12)]-dt[t+2^(12)]}

=2^(12)16In|[t-2^(12)][t+2^(12)]|+C

=2^(12)16In|[x^4-2^(12)][x^4+2^(12)]|+C

2.(x^3-1)(x^2+1)=x-x(x^2+1)-1(x^2+1)

原式=x^22-(12)In(x^2+1)-arctanx+C