If In=∫01 1−x5ndx,then 557I10I11 is equal to - Infinity Learn by Sri Chaitanya

A
$\frac{55}{56}$
B
$\frac{56}{55}$
C
$\frac{55}{54}$
D
$\frac{54}{55}$

Solution:

$\begin{array}{l} I_{11} = \int_{0}^{1} \underset{I}{\underset{⏟}{{(1 - x^{5})}^{11}}} \cdot 1 d x \\ {= {(1 - x^{5})}^{11} \cdot x]}_{0}^{1} + 11 \int_{0}^{1} {(1 - x^{5})}^{10} 5 x^{4} \cdot x d x \\ = 0 - 55 \int_{0}^{1} {(1 - x^{5})}^{10} (1 - x^{5} - 1) d x \\ = - 55 \int_{0}^{1} {(1 - x^{5})}^{11} d x + 55 I_{10} \end{array}$

or $56 I_{11} = 55 I_{10}$

or $\frac{I_{10}}{I_{11}} = \frac{56}{55}$

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