If $f(1+x) = f(1-x)$ ($\forall x \in \mathbb{R}$), then the value of the integral $I = \int_{-\infty}^{\infty} \frac{f(x)}{1+2^{(x-a)}} dx$ is
Step-by-Step Solution
Key Concept: Recognizing functional symmetry and applying the property of symmetric functions in definite integrals
As $f(1 + x) = f(1 - x)$, the function is symmetric about $x = 1$. Using the substitution property $(a + b - x)$, we get $2I = \int_0^2 f(t) + f(2-t)dt = I - 8$. Solving for $I$ gives $I = 8$.
Correct Answer: 8