Question Number 47743 by tanmay.chaudhury50@gmail.com last updated on 14/Nov/18
$$\int\frac{{dx}}{{x}^{{n}} −{x}}\:\: \\ $$$${the}\:{problems}\:{i}\:{have}\:{posted}\:{are}\:{tricky}… \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 14/Nov/18
$$\int\frac{{dx}}{{x}\left({x}^{{n}−\mathrm{1}} −\mathrm{1}\right)} \\ $$$$\int\frac{{x}^{{n}−\mathrm{2}} \:{dx}}{{x}^{{n}−\mathrm{1}} \left({x}^{{n}−\mathrm{1}} −\mathrm{1}\right)} \\ $$$${t}={x}^{{n}−\mathrm{1}} \:\:\:{dt}=\left({n}−\mathrm{1}\right){x}^{{n}−\mathrm{2}} {dx} \\ $$$$\int\frac{{dt}}{\left({n}−\mathrm{1}\right)\left({t}\right)\left({t}−\mathrm{1}\right)} \\ $$$$\frac{\mathrm{1}}{{n}−\mathrm{1}}\int\frac{{t}−\left({t}−\mathrm{1}\right)}{{t}\left({t}−\mathrm{1}\right)}{dt} \\ $$$$\frac{\mathrm{1}}{{n}−\mathrm{1}}\left[\int\frac{{dt}}{{t}−\mathrm{1}}−\int\frac{{dt}}{{t}}\right] \\ $$$$\frac{\mathrm{1}}{{n}−\mathrm{1}}\left[{ln}\left(\frac{{t}−\mathrm{1}}{{t}}\right)\right]+{c} \\ $$$$\frac{\mathrm{1}}{{n}−\mathrm{1}}\left[{ln}\left(\frac{{x}^{{n}−\mathrm{1}} −\mathrm{1}}{{x}^{{n}−\mathrm{1}} }\right)\right]+{c} \\ $$$$ \\ $$
Answered by ajfour last updated on 14/Nov/18
$${let}\:\:\:{t}=\frac{\mathrm{1}}{{x}}\:\:\:\Rightarrow\:\:{dt}\:=\:−\frac{{dx}}{{x}^{\mathrm{2}} } \\ $$$$\:\:\:\:{dx}\:=\:−\frac{{dt}}{{t}^{\mathrm{2}} } \\ $$$$\:\:{I}\:=\:\int\frac{\left(−\frac{{dt}}{{t}^{\mathrm{2}} }\right)}{\frac{\mathrm{1}}{{t}^{{n}} }−\frac{\mathrm{1}}{{t}}}\:=\:−\int\frac{{t}^{{n}−\mathrm{2}} {dt}}{\mathrm{1}−{t}^{{n}−\mathrm{1}} } \\ $$$$\:\:\:\:\:=\:\frac{\mathrm{1}}{{n}−\mathrm{1}}\int\:\frac{−\left({n}−\mathrm{1}\right){t}^{{n}−\mathrm{2}} {dt}}{\mathrm{1}−{t}^{{n}−\mathrm{1}} } \\ $$$$\:\:\:\:{I}\:=\:\frac{\mathrm{1}}{{n}−\mathrm{1}}\mathrm{ln}\:\mid\mathrm{1}−\frac{\mathrm{1}}{{t}^{{n}−\mathrm{1}} }\mid+{c}\:. \\ $$$$ \\ $$
Commented by rahul 19 last updated on 14/Nov/18
Nice
Is there any particular reason why u substitute t=1/x?
Commented by tanmay.chaudhury50@gmail.com last updated on 14/Nov/18
$${excellent}… \\ $$