The value of $k \in \mathrm{N}$ for which the integral $I_n=\int_0^1\left(1-x^k\right)^n d x, n \in…
The value of $k \in \mathrm{N}$ for which the integral $I_n=\int_0^1\left(1-x^k\right)^n d x, n \in \mathbb{N}$, satisfies $147 I_{20}=148 I_{21}$ is
14
8
10
7
Solution
$\begin{aligned} & I_n=\int_0^1\left(1-x^k\right)^n \cdot 1 d x \\ & I_n=\left(1-x^k\right)^n \cdot x-n k \int_0^1\left(1-x^k\right)^{n-1} \cdot x^{k-1} \cdot d x \\ & I_n=n k \int_0^1\left[\left(1-x^k\right)^n-\left(1-x^k\right)^{n-1}\right] d x \\ & I_n=n k I_n-n k I_n \\ & \frac{I_n}{I_{n-1}}=\frac{n k}{n k+1} \\ & \frac{I_{21}}{I_{20}}=\frac{21 k}{1+21 k} \\ & =\frac{147}{148} \Rightarrow k=7\end{aligned}$