求不定积分 ∫（dx/(x^3+1))
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∫（dx/(x^3+1))
=∫1/[(x+1)(x^2-x+1)] dx
=∫(1/3)/(x+1) + ((-1/3)x+(2/3))/(x^2-x+1) dx
=(1/3)ln|x+1|+∫(-1/3x+1/6)/(x^2-x+1) + (1/2)/(x^2-x+1) dx
=(1/3)ln|x+1| - (1/6)ln(x^2-x+1) + (1/2)∫1/[(x-(1/2))^2+4/3] dx
=(1/3)ln|x+1| - (1/6)ln(x^2-x+1) +(1/2)(2/√3)arctan((2x-1)/√3) + C
=(1/6)ln((x+1)^2/(x^2-x+1))+(1/√3)arctan((2x-1)/√3) + C
利用公式∫1/(x^2+a^2)dx=(1/a)arctan(x/a) + C
=∫1/[(x+1)(x^2-x+1)] dx
=∫(1/3)/(x+1) + ((-1/3)x+(2/3))/(x^2-x+1) dx
=(1/3)ln|x+1|+∫(-1/3x+1/6)/(x^2-x+1) + (1/2)/(x^2-x+1) dx
=(1/3)ln|x+1| - (1/6)ln(x^2-x+1) + (1/2)∫1/[(x-(1/2))^2+4/3] dx
=(1/3)ln|x+1| - (1/6)ln(x^2-x+1) +(1/2)(2/√3)arctan((2x-1)/√3) + C
=(1/6)ln((x+1)^2/(x^2-x+1))+(1/√3)arctan((2x-1)/√3) + C
利用公式∫1/(x^2+a^2)dx=(1/a)arctan(x/a) + C
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