The value of [k N] for which the integral [I_n=_0^1(1-x^k)^n d x, n N], satisfies 147 I_20…

Question

The value of $$k \in \mathbb{N}$$ 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

Accepted Answer

$$\begin{aligned}
& I(21)=\int_\limits0^1\left(1-x^k\right)^{21} d x \
& =\int_\limits0^1\left(1-x^k\right)\left(1-x^k\right)^{20} d x \
& =\int_\limits0^1\left(1-x^k\right)^{20} d x-\int_0 x^k\left(1-x^k\right)^{20} d x \
& I(21)=I(20)-\int_\limits0^1 x^k\left(1-x^k\right)^{20} d x
\end{aligned}$$

$$I(21)=I(20)-\left\lfloor\frac{\left(1-x^k\right)^{21}}{-21 k} x-\int_\limits0^1 \frac{(1-x^k)^{21}}{-21 k} d x\right\rfloor$$

$$\begin{aligned}
& I(21)=I(20)-\frac{1}{21 k} I(20) \
& \Rightarrow[I(21)](21 k+1)=21 K I(20) \
& \Rightarrow 21 K=147 \Rightarrow K=7
\end{aligned}$$

Solutions · 1