Question

lf $f\left(x\right)$ and $g\left(x\right)$ are any two continuous functions $\forall x\in R$ then ${\int }_{-a}^a\frac{g\left(x\right)-g(-x)}{f\left(x\right)+f(-x)}$ dx

• A. $0$
• B. $4$
• C. $6$
• D. $3$

Answer 1

The correct option is A $0$

Let $I={\int }_{-a}^a\frac{g\left(x\right)-g(-x)}{f\left(x\right)+f(-x)}dx$
Using ${\int }_a^{b}f\left(x\right)dx={\int }_a^{b}f(a+b-x)dx$
$I={\int }_{-a}^a\frac{g\left(-x\right)-g\left(x\right)}{f\left(-x\right)+f\left(x\right)}dx=-{\int }_{-a}^a\frac{g\left(x\right)-g(-x)}{f\left(x\right)+f(-x)}dx=-I\Rightarrow 2I=0\Rightarrow I=0$