Let ( m, n ) = 0 1 x m - 1 ( 1 - x ) n - 1 ? d x , m , n > 0 . If 0 1 ( 1 - x 10 ) 20 ? d x = a × ( b , c ), then 100 ( a + b + c ) equals ________. [2024]

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Solution

Q.
Let $β (m, n) = \int_{0}^{1} x^{m - 1} {(1 - x)}^{n - 1} d x, m, n > 0 .$

$If \int_{0}^{1} {(1 - x^{10})}^{20} d x = a \times β (b, c), then 100 (a + b + c) equals$ ________. [2024]

1 2012

2 2120

3 1021

4 1120

Ans.
(2)

Let $I = \int_{0}^{1} {(1 - x^{10})}^{20} d x$

Put $x^{10} = t$

$\Rightarrow x = t^{1 / 10}$

$\Rightarrow d x = \frac{1}{10} t^{- 9 / 10} d t$

$∴$ $I = \int_{0}^{1} {(1 - t)}^{20} \frac{1}{10} t^{- 9 / 10} d t = \frac{1}{10} \int_{0}^{1} t^{- 9 / 10} {(1 - t)}^{21 - 1} d t$

$= \frac{1}{10} β (\frac{1}{10}, 21) \Rightarrow a = \frac{1}{10}, b = \frac{1}{10}, c = 21$

$∴ 100 (a + b + c) = 100 (\frac{2}{10} + 21) = 2120$

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