Definite Integration
Grade 12
Question:
<p>The value of <span class="math-tex">\(\int_{0}^{1}\left(2 x^{3}-3 x^{2}-x+1\right)^{\frac{1}{3}} d x\)</span> is equal to</p>
<p style="display:inline">2</p>
<p style="display:inline">0</p>
<p style="display:inline">3</p>
<p style="display:inline">1</p>
Step-by-Step Solution
Key Concept: Recognize that f(x) = (2x³ - 3x² - x + 1)^(1/3) satisfies the property f(1-x) = -f(x) (odd symmetry about x = 1/2). For such functions, ∫₀¹ f(x)dx = 0 due to cancellation of positive and negative regions.
<p>Let <span class="math-tex">${I}=\int_{0}^{1}\left(2 x^{3}-3 x^{2}-x+1\right)^{\frac{1}{3}} d x$</span><br />
Using Property <span class="math-tex">$ f(x) d x$</span> where <span class="math-tex">$f(2 a- x)=-f(x)$</span><br />
Here <span class="math-tex">$f(1-x)=f(x)$</span><br />
<span class="math-tex">$\therefore I=0$</span></p>
Correct Answer: B