1. Tardigrade
  2. Question
  3. Mathematics
  4. For Un=∫ limits01 xn(2-x)n d x ; Vn=∫ limits01 xn(1-x)n d x n ∈ N, which of the following statement(s) is/are true?

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Q. For $U_n=\int\limits_0^1 x^n(2-x)^n d x ; V_n=\int\limits_0^1 x^n(1-x)^n d x n \in N$, which of the following statement(s) is/are true?

Integrals

Solution:

Given $U _{ n }=\int\limits_0^1 x ^{ n } \cdot(2- x )^{ n } dx ; V _{ n }=\int\limits_0^1 x ^{ n } \cdot(1- x )^{ n } dx$ in $U_n$
put $x=2 t \Rightarrow d x=2 d t$
$\therefore U_n=2 \int\limits_0^{1 / 2} 2^n \cdot t^n 2^n(1-t)^n d t$....(1)
Now $ V_n=2 \int\limits_0^{1 / 2} x^n(1-x)^n d x$(Using Queen)....(2)
From (1) and (2)
$U _{ n }=2^{2 n } \cdot V _{ n } \Rightarrow( C )$