Question Number 226339 by Spillover last updated on 25/Nov/25
Answered by Ghisom_ last updated on 26/Nov/25
![=∫(x^5 /(x^(12) −1))dx= [t=x^6 ] =(1/6)∫(dt/(t^2 −1))=(1/(12))ln ((t−1)/(t+1)) = =(1/(12))ln ((∣x^6 −1∣)/(x^6 +1)) +C](https://www.tinkutara.com/question/Q226349.png)
$$=\int\frac{{x}^{\mathrm{5}} }{{x}^{\mathrm{12}} −\mathrm{1}}{dx}= \\ $$$$\:\:\:\:\:\left[{t}={x}^{\mathrm{6}} \right] \\ $$$$=\frac{\mathrm{1}}{\mathrm{6}}\int\frac{{dt}}{{t}^{\mathrm{2}} −\mathrm{1}}=\frac{\mathrm{1}}{\mathrm{12}}\mathrm{ln}\:\frac{{t}−\mathrm{1}}{{t}+\mathrm{1}}\:= \\ $$$$=\frac{\mathrm{1}}{\mathrm{12}}\mathrm{ln}\:\frac{\mid{x}^{\mathrm{6}} −\mathrm{1}\mid}{{x}^{\mathrm{6}} +\mathrm{1}}\:+{C} \\ $$
Commented by Spillover last updated on 26/Nov/25
$${thank}\:{you} \\ $$