Definite Integration
Properties of Definite Integrals
Grade 12
Question:
<p>If <span>\(f(x) = \begin{vmatrix} \cos x & e^{x^2} & 2x\cos(x^2/2) \\ x^2 & \sec x & \sin x + x^3 \\ 1 & 2 & x + \tan x \end{vmatrix}\)</span>, the value of <span>\(\int_{-\pi/2}^{\pi/2} (x^2 + 1)[f(x) + f'(x)] dx\)</span> is</p>
<p>(a) -1</p>
<p>(b) 0</p>
<p>(c) 1</p>
<p>(d) 2</p>
Step-by-Step Solution
Key Concept: Check whether f(x) or components of the integrand are odd or even functions, and use symmetry to simplify integration over symmetric intervals.
<p>Analyze the symmetry properties of f(x). Expand the determinant and observe that the integrand has properties of odd/even functions over the symmetric interval <span>$[-\pi/2, \pi/2]$</span>. The result is 0.</p>
Correct Answer: B