Question

If $n (2 n +1) \int\limits_0^1\left(1- x ^{ n }\right)^{2 n } dx =1177 \int\limits_0^1\left(1- x ^{ n }\right)^{2 n +1} dx \text {, }$ then $n \in N$ is equal to _____

Answer 1

Let $I _1=\int\limits_0^1\left(1- x ^{ n }\right)^{2 n } dx$,
$I_2=\int\limits_0^1\left(1-x^n\right)^{2 n+1} d x$
$I_2=\int\limits_0^1\left(1-x^n\right)^{2 n+1} \cdot 1 d x$
$=\left.\left(1-x^n\right)^{2 n+1} \cdot x\right|_0 ^1-\int\limits_0^1(2 n+1)\left(1-x^n\right)^{2 n}\left(-n x^{n-1}\right) x d x$
$I_2=-n(2 n+1)\left\{I_2-I_1\right\} $
$\left(2 n^2+n+1\right) I_2=n(2 n+1) I_1 $
$\frac{I_1}{I_2}=\frac{2 n^2+ n +1}{ n (2 n +1)}=\frac{1177}{ n (2 n +1)} $
$\Rightarrow 2 n ^2+ n -1176=0 \Rightarrow n =24$