计算不定积分：∫dxx(x8+1)．

Question

计算不定积分：∫(dx)/(x(x8+1))．

Answer 1

∫(dx)/(x(x8+1))=∫(x7dx)/(x8(x8+1))＝(1)/(8)∫(dx8)/(x8(x8+1))

令x8＝t

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=(1)/(8)∫(dt)/(t(t+1))=(1)/(8)∫((1)/(t)−(1)/(t+1))dt=(1)/(8)ln|(t)/(t+1)|+c=−(1)/(8)ln(1+(1)/(x8))+c．