Let f and g be continuous functions on [0, a] such that f(x) = f(a -- x) and g(x) + g(a --…

Question

Let f and g be continuous functions on [0, a] such that f(x) = f(a -- x) and g(x) + g(a -- x) = 4, then $$\int\limits_0^a \, $$f(x) g(x) dx is equal to :

Accepted Answer

$${\rm I} = \int_0^a {f\left( x \right)g\left( x \right)dx} $$

$${\rm I} = \int_0^a {f\left( {a - x} \right)g\left( {a - x} \right)dx} $$

$${\rm I} = \int_0^a {f\left( x \right)\left( {4 - g\left( x \right.} \right)dx} $$

$${\rm I} = 4\int_0^a {f\left( x \right)dx - {\rm I}} $$

$$ \Rightarrow {\rm I} = 2\int_0^a {f\left( x \right)dx} $$

Solutions · 1