Let f and g be continuous functions on
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 $\int_{0}^{a} f(x) g(x) d x$ is equal to:

1. (1) $\quad 4 \int_{0}^{a} f(x) d x$

2. (2) $\int_{0}^{a} f(x) d x$

3. (3) $2 \int_{0}^{a} f(x) d x$

4. (4) $-3 \int_{0}^{a} f(x) d x$

Correct Option: , 3

Solution:

$f(x)=f(a-x)$

$g(x)+g(a-x)=4$

Let, the integral,

$I=\int_{0}^{a} f(x) g(x) d x$

$=\int_{0}^{a} f(a-x) \cdot g(a-x) d x$

$\left[\because \int_{a}^{b} f(x) d x=\int_{a}^{b} f(a+b-x) d x\right]$

$\Rightarrow \quad I=\int_{0}^{a} f(x)[4-g(x)] d x$

$\Rightarrow \quad I=\int_{0}^{a} 4 f(x) d x-\int_{0}^{a} f(x) \cdot g(x) d x$

$\Rightarrow \quad I=\int_{0}^{a} 4 f(x) d x-I$

$\Rightarrow \quad 2 I=\int_{0}^{a} 4 f(x) d x$

$\Rightarrow \quad I=2 \int_{0}^{a} f(x) d x$