The value of _0^1 [3] 2 x^3 - 3 x^2 - x + 1 ,dx is - Vidyadip

MCQ

The value of $\int\limits_0^1 {\sqrt[3]{{2{x^3} - 3{x^2} - x + 1}}\,dx} $ is

A
$-1$
✓
$0$
C
$1$
D
$2$

✓

Answer

Correct option: B.

$0$

b
Using $\int\limits_0^1 {f\left( x \right)} dx = \int\limits_0^1 {f\left( {1 - x} \right)dx} $

$I = \int\limits_0^1 {\sqrt[3]{{{x^2}(2x - 3) + (1 - x)}}} dx$

$ = \int\limits_0^1 {\sqrt[3]{{{{(1 - x)}^2}( - 1 - 2x) + x}}} dx$

$ = - \int\limits_0^1 {\sqrt[3]{{\left( {{x^2} - 2x + 1} \right)(1 + 2x) - x}}} dx$

$ = - \int\limits_0^1 {\sqrt {2{x^3} - 3{x^2} - x + 1} } dx = - I$

$2 \mathrm{I}=0 \quad \therefore \quad \mathrm{I}=0$

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