Question

Let f,g and h be continuous function on [0,a] such that

f(x) = f(a–x), g(x) = – g(a–x) and 3h(x) – 4h(a – x) = 5.

Then $\int_{0}^{a}{f(x)}$ g(x) h(x) dx =

• A. 5/4
• B. 3/4
• C. 1
• D. 0

Answer 1

Answer: (D)

0

Answer 2

$\int_{0}^{a}{f(x)}$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) $\left\lbrack \frac{3h(x) - 5}{4} \right\rbrack$dx

= – $\frac{3}{4}\int_{0}^{a}{f(x)}$g(x) h(x) dx + $\frac{5}{4}\int_{0}^{a}{f(x)}$g(x) dx

$\frac{7}{4}\int_{0}^{a}{f(x)}$ g(x) h(x) dx = $\frac{5}{4}\int_{0}^{a}{f(x)}$g(x) dx = 0

{f(a – x) g(a – x) = – f(x) g(x)}

So $\int_{0}^{a}{f(x)}$g(x) h(x) dx = 0