In the expansion of $\log[(1+x)^{1+x}.(1-x)^{1-x} ]$, what is the coefficient of $x^2$ ?
Solution
$\log [(1 + x)^{1+x}.(1-x)^{1-x}]$
$\log(1 + x)^{1+x} + \log(1 - x)^{1 - x}$
$(1 + x) \log (1 + x) + (1 - x) \log(1 - x)$
$(1 + x)\left[x - \dfrac{x^2}{2} + \dfrac{x^3}{3}.....\right] + (1 - x)\left[-x \dfrac{-x^2}{2}\dfrac{-x^2}{3}\right]$
Coefficient of $x^2$ is $+1$