Definite Integration
Integration with Composite Functions
Grade 12
Question:
<p>If <strong>f</strong>(<i>x</i>) = <math>∫</math><sup>x</sup><sub>0</sub> <i>g</i>(<i>t</i>) d<i>t</i>, where <i>g</i> is a non-zero even function. If <i>f</i>(<i>x</i> + 5) = <i>g</i>(<i>x</i>), then <math>∫</math><sup>0</sup><sub>0</sub> <i>f</i>(<i>t</i>) d<i>t</i> equals</p>
<p>(a) <math>5x + ∫<sup>x+5</sup><sub>5</sub> g(t) dt</math></p>
<p>(b) <math>∫<sup>x+5</sup><sub>5</sub> g(t) dt</math></p>
<p>(c) <math>2∫<sup>x+5</sup><sub>5</sub> g(t) dt</math></p>
<p>(d) <math>∫<sup>x+5</sup><sub>5</sub> g(t) dt</math></p>
Step-by-Step Solution
Key Concept: Use the given condition f(x+5) = g(x) and properties of even functions to transform and simplify the integral.
<p>This is a problem involving integration and composite functions. Using the given condition that f(x+5) = g(x) and the definition of f as an integral of an even function g, we apply integration by parts and properties of even functions to evaluate the definite integral.</p>
Correct Answer: B