When dealing with absolute values in integrals, there are a few approaches you can take. One method is to split the integral into different intervals based on the sign of the argument inside the absolute value. In this case, we can split the integral from -1 to 1 into two separate integrals: one from -1 to 0 and the other from 0 to 1. This allows us to remove the absolute value signs as follows:
\int^1_{-1} e^{-| |x| - \frac{1}{4} |} dx = \int^0_{-1} e^{-|x| + \frac{1}{4}} dx + \int^1_0 e^{|x| - \frac{1}{4}} dx
Next, we can use the fact that for any real number x, |x| = x when x is positive and |x| = -x when x is negative. This allows us to rewrite the integrands as follows:
\int^0_{-1} e^{-|x| + \frac{1}{4}} dx = \int^0_{-1} e^{-x + \frac{1}{4}} dx, and \int^1_0 e^{|x| - \frac{1}{4}} dx = \int^1_0 e^x dx
Now, we can simply evaluate each integral separately and add the results together to get the final answer. So, we have:
\int^0_{-1} e^{-x + \frac{1}{4}} dx = -e^{-x + \frac{1}{4}} \Big|^0_{-1} = -e^{\frac{1}{4}} + e^{\frac{5}{4}}
and \int^1_0 e^x dx = e^x \Big|^1_0 = e - 1
Therefore, the final answer is:
\int^1_{-1} e^{-| |x| - \frac{1}{4} |} dx = -e^{\frac{1}{4}} + e^{\frac{5}{4}} + e - 1
