Definite Integration
Basic Properties
Grade 12

Question:

<p>The integral \(\int_{1/2}^{1/2} \frac{[x]}{1-x} dx\) equals</p>
<p>(A) –1/2</p>
<p>(B) 0</p>
<p>(C) 1</p>
<p>(D) \(2\ln(1/2)\)</p>

Step-by-Step Solution

Key Concept: Any definite integral with equal upper and lower limits equals zero.
<p><strong>Solution:</strong> The integral is from 1/2 to 1/2, which is an integral over a zero-width interval.</p><p>By the definition of definite integration: $$\int_a^a f(x)dx = 0$$</p><p>Therefore: $$\int_{1/2}^{1/2} \frac{[x]}{1-x} dx = 0$$</p><p>∴ Answer is (B).</p>
Correct Answer: B

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