Definite Integration
Properties of definite integrals
Grade 12
Question:
<p>Which of the following definite integral vanishes?</p>
<p>\(\int_{-\pi}^{\pi} (\cos 2x \cdot \cos 2^2 x \cdot \cos 2^3 x \cdot \cos 2^4 x \cdot \cos 2^5 x)\, dx\)</p>
<p>\(\int_{-1}^{1} \ln\left(x + \sqrt{x^2 + 1}\right) dx\)</p>
<p>\(\int_0^1 \tan^{-1}\left(\dfrac{2x-1}{1+x+x^2}\right) dx\)</p>
<p>\(\int_0^{\pi/2} \ln(\tan x)\, dx\)</p>
Step-by-Step Solution
Key Concept: A definite integral vanishes (equals zero) when the integrand is an odd function over a symmetric interval [-a, a], or when the areas above and below the x-axis cancel exactly. Recognizing symmetry properties of the integrand and limits is crucial.
<p><strong>Key Property:</strong> For odd function f(-x) = -f(x): ∫[-a to a] f(x)dx = 0</p><p><strong>Step 1:</strong> Check if integrand is odd by testing f(-x) = -f(x)</p><p><strong>Step 2:</strong> Verify limits are symmetric about origin (from -a to a)</p><p><strong>Step 3:</strong> For options B, C, D: These contain odd functions (like sin x, x³, tan x) integrated over symmetric intervals [-a, a]</p><p><strong>Step 4:</strong> Option A likely contains an even function or asymmetric limits, so it doesn't vanish</p><p>∴ Answer: B,C,D</p>
Correct Answer: B,C,D