Indefinite integration

Question

Integrate this : $\int\frac{1}{1+(1+x)^\frac{1}{n}}dx$

I have tried it for many times but didn't get any easy solution for 10+2 level

Answer 1

Put $y=1+(1+x)^{1/n}$. Then $x=(y-1)^{n}-1$ and $dx=n(y-1)^{n-1} dy$. So the integral becomes $\int \frac 1 y n(y-1)^{n-1} dy$. To evaluate this expand $(y-1)^{n-1}$ using Binomial Theorem.