Definite Integration
Integration involving Derivatives
Grade 12
Question:
<p>If <span>\(f''(x) = k\)</span> in <span>\([0, a]\)</span>, then <span>\(\int_0^a \frac{f(x) - \frac{x^2}{2!} f''(x) - \frac{x^3}{3} xf(x)}{...} dx\)</span> is</p>
<p>(A) ka⁴/24</p>
<p>(B) ka²</p>
<p>(C) a³/3</p>
<p>(D) none of these</p>
Step-by-Step Solution
Key Concept: Use Taylor expansion and properties of higher-order derivatives in definite integration.
<p>Given <span>$f''(x) = k$</span>, integrate by parts and use Taylor expansion properties. The integral evaluates to <span>$\frac{ka^4}{24}$</span>.</p>
Correct Answer: A