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  • 1 # ForeveR72

    求1/(1+x^3)的不定積分

    1/(1+x^3)=1/(x+1)(x^2-x+1)=1/3[1/(x+1)-(x2)/(x^2-x+1)]

    ∫1/(1+x^3)dx=∫1/3[1/(x+1)-(x-2)/(x^2-x+1)]dx

    =∫1/3{1/(x+1)-[(x-1/2)+3/2]/[(x-1/2)^2+(√3/2)^2]}dx =∫1/3{1/(x+1)-[(x-1/2)+3/2]/[(x-1/2)^2+(√3/2)^2]dx

    =∫1/3{1/(x+1)-(x-1/2)/[(x-1/2

    )^2+(√3/2)^2]+3/2/[(x-1/2)^2+(√3/2)^2]}dx

    =1/3∫1/(x+1)dx-1/3∫(x-1/2)/[(x-1/2)^2+(√3/2)^2]dx+

    1/2∫/[(x-1/2)^2+(√3/2)^2]dx

    =1/3ln|x+1|-1/6ln|[(x-1/2)^2+(√3/2)^2]|+1/2*1/(√3/2)arctan[(x-1/2)/(√3/2)+C

    =1/3ln|x+1|-1/6ln|[(x-1/2)^2+(√3/2)^2]|+4/(3√3)*arctan[(x-1/2)/(√3/2)+C

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