Q96 / 150 · MHT CET 2019 (Shift 2) · Mathematics

$\int_{0}^{1} x {(1 - x)}^{5} d x = \dots \dots .$

A
$\frac{1}{5}$
Correct answer
B
$\frac{1}{42}$
C
$\frac{1}{13}$
D
$\frac{13}{42}$

Solution

Let =

l = \int_{0}^{1} x {(1 - x)}^{5} d x

Put

1 - x = t \Rightarrow x = 1 - t

\Rightarrow - d x = d t

\Rightarrow d x = - d t

When

x = 0^{'},

then

t = 1

and
When

x = 1, t h e n t = 0

∴ l \int_{1}^{0} (1 - t) {(t)}^{5} (- d t) = \int_{0}^{1} (t^{5} - t^{6}) d t

= (\frac{t^{6}}{6} - \frac{t^{7}}{7}) = \frac{1}{6} {(1)}^{6} - \frac{1}{7} {(1)}^{7}

= \frac{7 - 6}{42} = \frac{1}{42}

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

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