Definite Integration
Absolute Value Functions
Grade 12
Question:
<p>If <span class="math">\(\int_a^b |\sin x| dx = 8\)</span> and <span class="math">\(\int_{a+b}^b |\cos x| dx = 9\)</span>, then the value of <span class="math">\(\int_a^b \frac{x \sin x}{2x} dx\)</span> is:</p>
Step-by-Step Solution
Key Concept: The absolute value functions constrain the interval boundaries; use periodicity and properties of trigonometric integrals.
<p><strong>Solution:</strong> Use the given conditions about integrals of absolute trigonometric functions to determine the interval and evaluate the required integral.</p>
Correct Answer: 2