Homework help (Calc 2 - Nasty integral)

milehigh

So I'm a few weeks into Calc 2, and am doing some application problems. In any case I'm having a really hard time getting this integral as my calculator (and the back of the book) say it should come out...I've tried the few apparent u substitutions but am having no luck. Note that this isn't a required problem, it's just one I can't get and it's frustrating me.

I've got:

The integral from 1 to 8 of

sqrt((1+((4/9)x^(-2/3)) dx

The solution is stated as (1/27)(80(sqrt(10)-(13(sqrt)(13))

Any help or tips would be appreciated, I started with a U substitution of U=1+((4/9)x^(-2/3), but this leaves me up a creek later when I can't get everything in terms of U (namely dx) I've already put too much time into this, but I really want to know how to solve it, thanks in advance 8-)

Casually Hardcore

I haven't solve the problem. But I think I figured it out.

Since this is a homework problem, I'll be as vague as possible.

Anyways, try simplifying.

edit:

Okay, solved it. Yeah, you have to simplify.

Another hint, a lot of nasty integrals can be made to be quite easy.

milehigh

I really want to solve this integral just to have the ability to do it.

With that said I did manage to solve the problem by going back to the original equation and putting everything in terms of y rather than x, it made the resulting integral much easier. I do think I'll put some effort into solving it as shown however just for the practice though, any specific tips for problems of this type other than just simplifying?

Casually Hardcore

Recall your square root properties:

Sqrt(x/y) = sqrt(x)/sqrt(y)

sqrt(xy) = sqrt(x)sqrt(y)

also recall your exponential properties (which kinda overlaps with the sqrt properties, but I'm no math major).

(a^x)^y = a^xy

now, also recall that

a+ a/b = (ab +a)/b

Now try simplifying the original problem before you integrate. Eventually you'll get something that you can u-sub.