Question

The value of ${\int }_0^{12a}\left[\frac{f(x)}{f\left(x\right)+f(12a-x)}\right]dx$ is

• A. $a$
• B. $2a$
• C. $3a$
• D. $6a$

Answer 1

The correct option is D

$6a$

Explanation for the correct answer.

Evaluate the integral.

$$\begin{gathered} I={\int }_0^{12a}\left[\frac{f(x)}{f\left(x\right)+f(12a-x)}\right]dx \\ \Rightarrow I={\int }_0^{12a}\left[\frac{f(12a-x)}{f(12a-x)+f\left(x\right)}\right]dx \\ \Rightarrow 2I={\int }_0^{12a}\left[\frac{f(x)+f(12a-x)}{f(12a-x)+f\left(x\right)}\right]dx \\ \Rightarrow 2I={\int }_0^{12a}dx \\ ={\left[x\right]}_0^{12a} \\ =(12a-0) \\ =12a \\ \therefore I=6a \end{gathered}$$

Hence, option D is correct .