How do you deal with absolute values in your Integrals?

[Shinjo]

I have to solve the integral:

$$\int^1_{-1} e^{-| |x| - \frac{1}{4} |} dx$$

but I have no idea what to do with the absolute value signs. Can someone help me?

 

[Data]

When is $$x < 0$$? When is $$|x|-1/4 <0$$? Split up the integral as is appropriate. For example,

$$\int_{-5}^5 e^{|x|} \ dx = 2\int_0^5 e^x \ dx$$

$$\int_0^5 |x-4| \ dx = \int_0^4 4-x \ dx + \int_4^5 x-4 \ dx$$

 

[M98Ranger]

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