Integral $\int \frac{1}{x(1-x)^n}dx$

Question

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?

Answer 1

To include non-integer $n$, Maple uses a hypergeometric $$ \int \frac{dx}{x(1-x)^n} = nx\;\mbox{$_3$F$_2$}\big(1,1,1+n;\,2,2;\,x\big)+\ln \left( x \right) + C $$

Answer 2

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.