I am try to integrate the following function:$$\frac{1}{x(1-x)^n}$$ I have searched on some online integral calculator, but they are not solving it. Can we use binomial theorem in this?
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$\begingroup$Not sure about the Binomial theorem, but if you make substitution $y=1-x$, it becomes $-\int \frac{1}{(1-y)y^n}$. Now notice that $$ \frac{1}{(1-y)y^n} = \frac{1}{y^n}+\frac{1}{y^{n-1}}+\dots+\frac{1}{y}+\frac{1}{1-y} $$(you might want to think why this is). So you have finitely many terms you should be able to integrate.
$\endgroup$ $\begingroup$To include non-integer $n$, Maple uses a hypergeometric$$ \int \frac{dx}{x(1-x)^n} = nx\;\mbox{$_3$F$_2$}\big(1,1,1+n;\,2,2;\,x\big)+\ln \left( x \right) + C $$
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