Definite Integration
Integration of Determinants with Trigonometric Functions
Grade 12
Question:
<p>If <span>\(f(x) = \begin{vmatrix} \sin x + \cos x & \ln(\cos x) & 1 + (\tan x)^2 \\ \pi & \pi^2 & \pi^4 \\ 1 & 7 & 1 \end{vmatrix}\)</span>, the value of <span>\(\int_0^{\pi/2} f(x) dx\)</span> is</p>
<p>(a) not defined</p>
<p>(b) 0</p>
<p>(c) 1</p>
<p>(d) undefined</p>
Step-by-Step Solution
Key Concept: When a determinant has a row of constants, the determinant value may simplify significantly, and properties of the trigonometric and logarithmic terms in the first row determine the result.
<p>Note that the second row contains only constants (π, π², π⁴). Evaluate the determinant by expanding along the second row or analyzing the structure. The integral depends on the behavior of the first row's functions.</p>
Correct Answer: B