1、∫ 3/(x³+1) dx =∫ 3/[(x+1)(x²-x+1)] dx 令3/[(x+1)(x²-x+1)]=A/(x+1)+(Bx+C)/(x²-x+1) 右邊通分相加與左邊比較係數，得：A=1，B=-1，C=2 =∫ 1/(x+1) dx - ∫ (x-2)/(x²-x+1) dx =ln|x+1| - (1/2)∫ (2x-1-3)/(x²-x+1) dx =ln|x+1| - (1/2)∫ (2x-1)/(x²-x+1) dx + (1/2)∫ 3/(x²-x+1) dx =ln|x+1| - (1/2)∫ 1/(x²-x+1) d(x²-x) + (3/2)∫ 1/[(x-1/2)²+3/4] dx =ln|x+1| - (1/2)ln(x²-x+1) + √3arctan[(2x-1)/√3] + C 2、∫ 1/[x(x^6+4)] dx 分子分母同乘以x^5 =∫ x^5/[x^6(x^6+4)] dx =(1/6)∫ 1/[x^6(x^6+4)] d(x^6) =(1/24)∫ [1/x^6 - 1/(x^6+4)] d(x^6) =(1/24)[lnx^6 - ln(x^6+4)] + C =(1/4)ln|x| - (1/24)ln(x^6+4) + C 【數學之美】團隊為您解答，若有不懂請追問，如果解決問題請點下面的“選為滿意答案”。