Is My Book Correct About the Integral of e^-ax from Negative Infinity to 0?

[leopard]

Is my book right to write

\int ^{0}_{- \infty}e^{-a|x|}dx = \int ^{0}_{- \infty}e^{ax}dx

?

In case, why?

 

[mutton]

If x is negative, what is |x|?

 

[leopard]

positive, but -a is still negative.

 

[HallsofIvy]

That was not the question. If x is negative, then |x|= -x. For x negative, -a|x|= (-a)(-x)= ax.

 

[gabbagabbahey]

errmmm...maybe it's best to think of an example...suppose you have x=-2, what is |x|?

 

[leopard]

|-2| = 2

 

[gabbagabbahey]

|-2| = 2

Right, and if x=-2, what is -x?

 

[leopard]

Should be 2. I can simply put the minus outside the brackets?

 

[gabbagabbahey]

Should be 2. I can simply put the minus outside the brackets?

Well, if |x|=2 and -x=2, then surely you can say that |x|=-x?


So...what does that make -a|x|?