Integral 1/x^(2/3)dx from -1 to 1: Solution

[miglo]

Homework Statement

integral 1/x^(2/3)dx from -1 to 1

Homework Equations

The Attempt at a Solution

so i split it up into two integrals, one with limits going from -1 to b and the other with limits going from c to 1, and taking the limits as b and c go to 0
i know my antiderivative is 3x^(1/3) and i plugged in my limits of integration and then took the limit as both b and c went to 0 and i got an answer of 3-3(-1)^(1/3) i know this is wrong simply because the cube root of -1 will be give me a complex number but i tried it on wolfram alpha and it gave me the same answer. i looked at the back of my book and it says the answer is 6, i don't understand how the book got an answer of 6

 

[Dick]

Wolfram Alpha is wrong and so is plugging into the antiderivative (which it's likely WA did) for an improper integral. 1/x^(2/3) is actually an even function if you take it to be (1/x^(1/3))^2, isn't it? That should mean you can take the improper integral over (0,1] and double it. What's that integral? BTW (-1)^(1/3) is not necessarily a complex number.

 

[miglo]

ohh thanks! ill watch out for even and odd functions

one more quick question though
i have another improper integral, its 2dx/(x^2-1) from -infinity to -2
so i broke it up into two integrals, the first one has limits from a to -1 with the limit as a approaches -infinity and the second is from -1 to -2
am i doing this right?

 

[Dick]

ohh thanks! ill watch out for even and odd functions

one more quick question though
i have another improper integral, its 2dx/(x^2-1) from -infinity to -2
so i broke it up into two integrals, the first one has limits from a to -1 with the limit as a approaches -infinity and the second is from -1 to -2
am i doing this right?

Why do you need the integral from -1 to -2? -1 isn't in (-infinity,-2). There's no need to break it there. Just integrate from a to -2.

 

[miglo]

ohh alright
thanks again