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设 x = u^4,则 dx = 4u³ * du。那么,上式积分变换为:
=∫4u³ * du/(u + u²)
=4∫u² * du/(1+u)
=4∫[(u²-1) + 1] *du/(u+1)
=4 * [∫(u-1)du + ∫du/(u+1)]
=4 * [∫u * du - ∫du + ∫d(u+1)/(u+1)]
=2 * ∫2u * du - 4 * ∫du + 4 * ln(u+1)
=2u² - 4u + 4 * ln(u+1) + C
=2 *(√x) - 4 * x^(1/4) + 4 * ln[x^(1/4) + 1] + C
=∫4u³ * du/(u + u²)
=4∫u² * du/(1+u)
=4∫[(u²-1) + 1] *du/(u+1)
=4 * [∫(u-1)du + ∫du/(u+1)]
=4 * [∫u * du - ∫du + ∫d(u+1)/(u+1)]
=2 * ∫2u * du - 4 * ∫du + 4 * ln(u+1)
=2u² - 4u + 4 * ln(u+1) + C
=2 *(√x) - 4 * x^(1/4) + 4 * ln[x^(1/4) + 1] + C
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