求对∫1/(x^(1/2)+x^(5/6))d x - 知识问答

求对∫1/(x^(1/2)+x^(5/6))d x的不定积分

1个回答

1/2 和 5/6 的公因数是 1/6
设t = ⁶√x,x = t⁶,dx = 6t⁵ dt
∫ dx/[√x + x^(5/6)]
= ∫ (6t⁵)/(t³ + t⁵)
= 6∫ t²/(1 + t²) dt
= 6∫ (t² + 1 - 1)/(1 + t²) dt
= 6∫[1 - 1/(1 + t²)] dt
= 6t - 6arctan(t) + C
= 6(⁶√x) - 6arctan(⁶√x) + C

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