The value of $\int\limits_{-a}^{a}f(x)dx$ where $f(x) =\frac{7^x}{1+7^x}$, is:

$a$

The correct answer is Option (1) → $a$ **

Given:

$f(x)=\frac{7^{x}}{1+7^{x}}$

Compute $f(-x)$:

$f(-x)=\frac{7^{-x}}{1+7^{-x}}=\frac{1}{7^{x}+1}$

Now add:

$f(x)+f(-x) =\frac{7^{x}}{1+7^{x}}+\frac{1}{1+7^{x}} =\frac{7^{x}+1}{1+7^{x}} =1$

Thus:

$\displaystyle \int_{-a}^{a} f(x)\,dx =\int_{0}^{a}\bigl(f(x)+f(-x)\bigr)\,dx =\int_{0}^{a} 1\,dx =a$

Final Answer: $a$