求不定积分∫(x^4+1)/(x^6+1)dx

Question

求不定积分 ∫ (x^4+1)/(x^6+1)dx

Answer 1

∫ (x^4+1)/(x^6+1)dx
=∫ (x^4+1-x^2+x^2)/(x^6+1)dx
=∫ (x^4+1-x^2)/(x^6+1)dx+∫ x^2/(x^6+1)dx
注意到：(x^4+1-x^2)(x^2+1)=x^6+1
=∫ 1/(x^2+1)dx+1/3∫ 1/(x^6+1)d(x^3)
=arctanx+1/3arctan(x^3)+C