The value of ∫ (2x^3 − 3x^2 − x + 1)^1/3 dx; x ∈ [0, 1] is equal to:

Question

The value of \(\int\limits_{0}^{1}\left(2 x^{3}-3 x^{2}-x+1\right)^{\frac{1}{3}} d x \) is equal to:

(1) 0

(2) 1

(3) 2

(4) -1

Answer 1

Correct option is (1) 0

\(I=\int\limits_0^1\left(2 x^3-3 x^2-x+1\right)^{1 / 3} \cdot d x \)

Applying king replace \(x \rightarrow(1-x) \)

\( I=\int\limits_0^1\left[2(1-x)^3-3(1-x)^2-(1-x)+1\right]^{1 / 3} \cdot d x\)

\( I=\int\limits_0^1\left[2-2 x^3-6 x+6 x^2-3-3 x^2+6 x-1+x+1\right]^{1 / 3} \cdot d x\)

\(I=\int\limits_0^1\left[-2 x^3+3 x^2+x-1\right]^{1 / 3} \cdot d x\)

\(I=-\int\limits_0^1\left[2 x^3-3 x^2-x+1\right]^{1 / 3} \cdot d x\)

\(I=-I\)

\(2 I=0 \Rightarrow I=0\)