求不定积分 (1-√x-1)/1+三次根号下x-1
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求不定积分 (1-√x-1)/1+三次根号下x-1
![求不定积分 (1-√x-1)/1+三次根号下x-1](/uploads/image/z/2773911-39-1.jpg?t=%E6%B1%82%E4%B8%8D%E5%AE%9A%E7%A7%AF%E5%88%86+%281-%E2%88%9Ax-1%29%2F1%2B%E4%B8%89%E6%AC%A1%E6%A0%B9%E5%8F%B7%E4%B8%8Bx-1)
设 (x-1)^(1/6)=t,则 x=1+t^6,dx=6t^5;
∫[1-√(x-1)]/[1+(x-1)^(¹/³)]dx=∫[(1-t³)/(1+t²)]*(6t^5)dt=6∫(1-t-t²+t³ +t^4 -t^6)+[(t-1)/(1+t²)] dt
=6t-3t²-2t³+(3t^4 /2)-(6t^7 /7)+6∫[(t-1)/(1+t²)]dt
=6t-3t²-2t³+(3t^4 /2)-(6t^7 /7)+3ln(1+t²)-6arctant+C;
将 t 重新换为 (x-1)^(¹/6) 即可:
=6(x-1)^(¹/6)-3(x-1)^(¹/³)-2√(x-1) +(3/2)(x-1)^(²/³)-(6/7)(x-1)(x-1)^(¹/6)+3ln[1+(x-1)^(¹/³)]-6arctan[(x-1)^(¹/6)]+C;
∫[1-√(x-1)]/[1+(x-1)^(¹/³)]dx=∫[(1-t³)/(1+t²)]*(6t^5)dt=6∫(1-t-t²+t³ +t^4 -t^6)+[(t-1)/(1+t²)] dt
=6t-3t²-2t³+(3t^4 /2)-(6t^7 /7)+6∫[(t-1)/(1+t²)]dt
=6t-3t²-2t³+(3t^4 /2)-(6t^7 /7)+3ln(1+t²)-6arctant+C;
将 t 重新换为 (x-1)^(¹/6) 即可:
=6(x-1)^(¹/6)-3(x-1)^(¹/³)-2√(x-1) +(3/2)(x-1)^(²/³)-(6/7)(x-1)(x-1)^(¹/6)+3ln[1+(x-1)^(¹/³)]-6arctan[(x-1)^(¹/6)]+C;