The value of 0 1 ( 2 x 3 - 3 x 2 - x + 1 ) 1 3 ? d x is equal to : [2024]

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Solution

Q.
The value of $\int_{0}^{1} {(2 x^{3} - 3 x^{2} - x + 1)}^{\frac{1}{3}} d x$ is equal to : [2024]

1 - 1

2 2

3 0

4 1

Ans.
(3)

Let $I = \int_{0}^{1} {(2 x^{3} - 3 x^{2} - x + 1)}^{1 / 3} d x$

$\Rightarrow I = \int_{0}^{1} {[2 {(1 - x)}^{3} - 3 {(1 - x)}^{2} - (1 - x) + 1]}^{1 / 3} d x$

$\Rightarrow I = \int_{0}^{1} (- 2 x^{3} + 3 x^{2} + x - 1) d x$

$\Rightarrow I = \int_{0}^{1} - (2 x^{3} - 3 x^{2} - x + 1) d x$

$\Rightarrow I = - I \Rightarrow 2 I = 0 \Rightarrow I = 0$

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