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Reduction formula for the integral of (1 - x^k)^n from 0 to 1

This type of question, asked 3 times: 26 Jul 2022, 6 Apr 2024, 8 Apr 2024. All 3 questions below, oldest first.

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  1. Question 1

    26 Jul 2022

    86
    If n(2n+1)∫01(1−xn)2ndx=1177∫01(1−xn)2n+1dx,n(2n + 1)\int_{0}^{1} \left( 1 -x^{n} \right)^{2n}dx = 1177\int_{0}^{1} \left( 1 -x^{n} \right)^{2n + 1}dx, then n∈Nn \in N is equal to

  2. Question 2

    6 Apr 2024

    83
    Let rk=∫01(1−x7)kdx∫01(1−x7)k+1dx,k∈Nr_{k} = \frac{\int_{0}^{1} \left( 1 - x^{7} \right)^{k}dx}{\int_{0}^{1} \left( 1 - x^{7} \right)^{k + 1}dx},k \in N. Then the value of ∑k=11017(rk−1)\sum_{k = 1}^{10} \frac{1}{7\left( r_{k} - 1 \right)} is equal to .

  3. Question 3

    8 Apr 2024

    61
    The value of k∈Nk \in\mathbb{N} for which the integral
    In=∫01(1−xk)ndx,n∈N, satisfies 147I20=148I21I_{n}=\int_{0}^{1} \left( 1 -x^{k} \right)^{n}dx,n\in\mathbb{N}\text{, satisfies~}147I_{20}= 148I_{21}
    is:

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