Definite Integration
Even/odd function properties
Grade 12
Question:
<p>\(\int_{-\pi/2}^{-\pi/2} [(x+\pi)^3 + \cos^2(x+3\pi)]\, dx\) is equal to</p>
<p>\(\dfrac{\pi^4}{32}\)</p>
<p>\(\dfrac{\pi^4}{32} + \dfrac{\pi}{2}\)</p>
<p>\(\dfrac{\pi}{2}\)</p>
<p>\(\dfrac{\pi}{4} - 1\)</p>
Step-by-Step Solution
Key Concept: The integral has identical upper and lower limits (-π/2), making the integration interval length zero. Any definite integral over a zero-length interval equals zero, regardless of the integrand's complexity.
<p><strong>Step 1:</strong> Identify the limits of integration.</p><p>Upper limit = -π/2</p><p>Lower limit = -π/2</p><p><strong>Step 2:</strong> Apply the fundamental property of definite integrals.</p><p>When the upper limit equals the lower limit, we have:</p><p>$$\int_{a}^{a} f(x)\,dx = 0$$</p><p><strong>Step 3:</strong> Conclusion.</p><p>Since both limits are -π/2, regardless of what the integrand is:</p><p>$$\int_{-\pi/2}^{-\pi/2} [(x+\pi)^3 + \cos^2(x+3\pi)]\,dx = 0$$</p><p><strong>∴ Answer: 0</strong></p>
Correct Answer: C