Definite Integration
General
Grade 12
Question:
<p>If $f, g, h$ be continuous functions on $[0, a]$ such that $f(a - x) = -f(x)$, $g(a - x) = g(x)$ and $3h(x) - 4h(a - x) = 5$, then prove that $\int_{0}^{a} f(x)g(x)h(x) dx = 0$</p>
Step-by-Step Solution
Key Concept: General
<div>$I = \int_{0}^{a} f(x)g(x)h(x) dx = \int_{0}^{a} f(a - x)g(a - x)h(a - x) dx = -\int_{0}^{a} f(x)g(x)h(a - x) dx$<br>$7I = 3I + 4I = \int_{0}^{a} f(x)g(x) \{3h(x) - 4h(a - x)\} dx = 5 \int_{0}^{a} f(x)g(x) dx = 0$<br>(since $f(a - x)g(a - x) = -f(x)g(x)$) $\Rightarrow I = 0$</div>
Correct Answer: A