Solving Integral: $\int_{-\infty}^{\infty} e^{-|x|} dx$

[G01]

\int_{-\infty}^{\infty} e^{-|x|} dx

Could someone tell me why this integral, when you split it comes out to be:

\int_{-\infty}^0 e^x dx + \int_0^{\infty} e^{-x} dx

I keep thinking it should be e^(-x) in the first integral. I don't know why its positive. I can solve this integral otherwise. Thanks again.

 

[nocturnal]

The absolute value is defined for a real number x by
<br /> |x|= \left \lbrace<br /> \begin{array}{l l}<br /> x &amp; \mbox{if} \ x \geq 0 \\<br /> -x &amp; \mbox{if} \ x\leq 0<br /> \end{array}<br /> \right.<br />

when x is in the interval (-\infty,0), then |x| = -x. Likewise, when x is in the interval (0,\infty), then |x| = x. Therefore,

\int_{-\infty}^0 e^{-|x|}dx = \int_{-\infty}^0 e^{-(-x)}dx<br /> = \int_{-\infty}^0 e^x dx

 

[G01]

Im sorry that definition confuses me... I always thought the absolute value was the positive distance of that number from the origin, leading me to believe that |-x|= x. This seems contrary.

 

[nocturnal]

Im sorry that definition confuses me... I always thought the absolute value was the positive distance of that number from the origin

This is true.

leading me to believe that |-x|= x. This seems contrary.

This is false. For example take x=-3. Then your saying that |-(-3)| = -3, which is the same as saying that |3| = -3, which is clearly false. I used paranthenses to delimit the value which was substituted for x, namely -3.

However, if we use the definition of absolute value, we arive at the correct result. For example, if x=-3, we have that x&lt;0 so we use the second case of the definition to arrive that |x| = -x, which gives |-3| = -(-3) = 3.

Try plugging in different values for x into the definition to convince yourself that this definition works.

The definition for when x&gt;0 should be clear. The absolute value of a positive number is a positive quantity and is equal to that number. However if x&lt;0, then -x&gt;0, so -x is a positive number (don't let the negative sign confuse you, remember that the negative of a negative number is positive).