Definite Integration
Grade 12
Question:
<p>Let <span class="math-tex">\(f(x)=\int_{0}^{x} g(t) d t\)</span>, where g is non-zero even function. If f (x + 5) = g (x), then <span class="math-tex">\(\int_{0}^{x} f(t) d t\)</span> equals</p>
<p style="display:inline"><span class="math-tex">\(\int\limits_{5}^{x+5} g(t) d t\)</span></p>
<p style="display:inline"><span class="math-tex">\(2 \int\limits_{5}^{x+5} g(t) d t\)</span></p>
<p style="display:inline"><span class="math-tex">\(\int\limits_{x+5}^{5} g(t) d t\)</span></p>
<p style="display:inline"><span class="math-tex">\(5 \int\limits_{x+5}^{5} g(t) d t\)</span></p>
Step-by-Step Solution
Key Concept: Use the Fundamental Theorem of Calculus to identify f'(x) = g(x), then substitute f(t) = f'(t-5) into the integral while applying the parity properties of f and g to simplify the bounds.
<p><span class="math-tex">$\int\limits_{x+5}^{5} g(t) d t$</span></p>
Correct Answer: C