^1_0 x (1-x)^ 54 dx= - Vidyadip

MCQ

$\int\limits^1_0\frac{\text{x}}{(1-\text{x})^{54}}\text{ dx}=$

A
$\frac{15}{16}$
B
$\frac{3}{16}$
C
$-\frac{3}{16}$
✓
$-\frac{16}{3}$

✓

Answer

Correct option: D.

$-\frac{16}{3}$

$\text{I}=\int\limits^1_0\frac{\text{x}}{(1-\text{x})^{\frac{5}{4}}}\text{ dx}$
Put$, 1 - x = t$
$\Rightarrow x =1 -t$
$\Rightarrow dx = -dt$

$x$	$0$	$1$
$t$	$1$	$0$

$\text{I}=\int\limits^0_1\frac{(1-\text{t})(-\text{dt})}{\text{t}^{\frac{5}{4}}}$
$\text{I}=\int\limits^1_0\Big(\text{t}^\frac{5}{4}-\text{t}^\frac{-1}{4}\Big)\text{dt}$
$\text{I}=\Bigg[\frac{\text{t}^{-\frac{5}{4}}}{\frac{-1}{4}}-\frac{\text{t}^\frac{3}{4}}{\frac{3}{4}}\Bigg]^1_0$
$\text{I}=-4-\frac{4}{3}$
$\text{I}=\frac{-16}{3}$

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