求不定积分(1-x^8)/(x+x9),(1-x^8)/(x+x^9)是这个

Question

求不定积分(1-x^8)/(x+x9) ,
(1-x^8)/(x+x^9)是这个

Answer 1

∫[(1-x^8)/(x+x^9)]dx=∫[(x^7)(1/x^8-1)/(1+x^8)]dx=(1/8)∫[(1/x^8-1)/(1+x^8)]dx^8
令t=1+x^8,则原积分化为(1/8)∫{[1/(t-1)-1]/t}dt=(1/8)∫{(2-t)/[t(t-1)]}dt=(1/8)∫[1/(t-1)]dt-(1/4)∫(1/t)dt=(1/8)ln(|t-1|/t^2)+C
将t代回x得原积分=ln|x|-(1/4)ln(1+x^8)+C