If ∫01x^m(1-x)^ndx=k∫01x^n(1-x)^mdx, then the value of k equals

Correct answer: (D) 1. Given: ∫01 1-x^nx^mdx=k∫01 x^n 1-x^mdx ⇒I(m,n)=∫01 x^m 1-x^ndx ⇒∫0a fxdx=∫0afa-xdx ⇒I(m,n)=∫01 1-x^m 1-1-x^ndx ⇒I(m,n)=∫01 1-x^m 1-1+x^ndx ⇒I(m,n)=∫01 1-x^m x^ndx ⇒k=1

Q119 / 160 · AP EAMCET 2021 (20 Aug Shift 2) · Mathematics

If $\int_{0}^{1} x^{m} (1 - x)^{n} d x = k \int_{0}^{1} x^{n} (1 - x)^{m} d x$ , then the value of $k$ equals

A
$m$
B
$n$
C
$\frac{1}{m n}$
Correct answer
D
$1$

Solution

$G i v e n : \int_{0}^{1} {(1 - x)}^{n} x^{m} d x = k \int_{0}^{1} x^{n} {(1 - x)}^{m} d x$

$\Rightarrow I (m, n) = \int_{0}^{1} x^{m} {(1 - x)}^{n} d x$

$\Rightarrow \int_{0}^{a} f (x) d x = \int_{0}^{a} f (a - x) d x$

$\Rightarrow I (m, n) = \int_{0}^{1} {(1 - x)}^{m} {[1 - (1 - x)]}^{n} d x$

$\Rightarrow I (m, n) = \int_{0}^{1} {(1 - x)}^{m} {[1 - 1 + x]}^{n} d x$

$\Rightarrow I (m, n) = \int_{0}^{1} {(1 - x)}^{m} x^{n} d x$

$\Rightarrow k = 1$

Hence, the correct answer is option: 'D'.

If ∫01xm(1-x)ndx=k∫01xn(1-x)mdx, then the value of k equals

Solution

Relevant Questions

If $\int_{0}^{1} x^{m} (1 - x)^{n} d x = k \int_{0}^{1} x^{n} (1 - x)^{m} d x$ , then the value of $k$ equals