求微积分的反导数,∫(x^3)+1/((x^3)-1) dx 如果看不清楚的,意思是求(x^3)+1除以(x^3)-1的
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求微积分的反导数,
∫(x^3)+1/((x^3)-1) dx 如果看不清楚的,意思是求(x^3)+1除以(x^3)-1的反导数.
∫(x^3)+1/((x^3)-1) dx 如果看不清楚的,意思是求(x^3)+1除以(x^3)-1的反导数.
![求微积分的反导数,∫(x^3)+1/((x^3)-1) dx 如果看不清楚的,意思是求(x^3)+1除以(x^3)-1的](/uploads/image/z/4365941-5-1.jpg?t=%E6%B1%82%E5%BE%AE%E7%A7%AF%E5%88%86%E7%9A%84%E5%8F%8D%E5%AF%BC%E6%95%B0%2C%E2%88%AB%28x%5E3%29%2B1%2F%28%28x%5E3%29-1%29+dx+%E5%A6%82%E6%9E%9C%E7%9C%8B%E4%B8%8D%E6%B8%85%E6%A5%9A%E7%9A%84%2C%E6%84%8F%E6%80%9D%E6%98%AF%E6%B1%82%28x%5E3%29%2B1%E9%99%A4%E4%BB%A5%28x%5E3%29-1%E7%9A%84)
这题不能不复杂吧,有难度.那我隔行写容易看:
∫ (x³+1) / (x³-1) dx
= ∫ (x³+1) / (x-1)(x²+x+1) dx
= ∫ [-2(x+2) / 3(x²+x+1) + 2 / 3(x-1) + 1] dx
= (-2/3)∫ (x+2)/(x²+x+1) dx + (2/3)∫ dx/(x-1) + ∫ dx
= (-2/3)∫ [(2x+1) / 2(x²+x+1) + 3 / 2(x²+x+1)] dx + (2/3)∫ dx/(x-1) + ∫ dx
= (-1/3)∫ (2x+1)dx/(x²+x+1) - ∫ dx/(x²+x+1) + (2/3)∫ dx/(x-1) + ∫ dx
= (-1/3)∫ (2x+1)dx/(x²+x+1) - ∫ dx/[(x+1/2)²+3/4] + (2/3)∫ dx/(x-1) + ∫ dx
= (-1/3)∫ d(x²+x+1)/(x²+x+1) - ∫ d(x+1/2)/[(x+1/2)²+3/4] + (2/3)∫ d(x-1)/(x-1) + ∫ dx
= (-1/3)ln|x²+x+1| - √(4/3)*arctan[√(4/3) * (x+1/2)] + (2/3)ln|x-1| + x + C
= (-1/3)ln|x²+x+1| + (2/3)ln|x-1| + x - (2/√3)arctan[(2x+1)/√3] + C
已经是最简易的答案了.
∫ (x³+1) / (x³-1) dx
= ∫ (x³+1) / (x-1)(x²+x+1) dx
= ∫ [-2(x+2) / 3(x²+x+1) + 2 / 3(x-1) + 1] dx
= (-2/3)∫ (x+2)/(x²+x+1) dx + (2/3)∫ dx/(x-1) + ∫ dx
= (-2/3)∫ [(2x+1) / 2(x²+x+1) + 3 / 2(x²+x+1)] dx + (2/3)∫ dx/(x-1) + ∫ dx
= (-1/3)∫ (2x+1)dx/(x²+x+1) - ∫ dx/(x²+x+1) + (2/3)∫ dx/(x-1) + ∫ dx
= (-1/3)∫ (2x+1)dx/(x²+x+1) - ∫ dx/[(x+1/2)²+3/4] + (2/3)∫ dx/(x-1) + ∫ dx
= (-1/3)∫ d(x²+x+1)/(x²+x+1) - ∫ d(x+1/2)/[(x+1/2)²+3/4] + (2/3)∫ d(x-1)/(x-1) + ∫ dx
= (-1/3)ln|x²+x+1| - √(4/3)*arctan[√(4/3) * (x+1/2)] + (2/3)ln|x-1| + x + C
= (-1/3)ln|x²+x+1| + (2/3)ln|x-1| + x - (2/√3)arctan[(2x+1)/√3] + C
已经是最简易的答案了.
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