# How to Integrate (x) sq. root of ((x^2) - 1)?

• Prototype
In summary: No worries, Jameson. Thanks for catching it! In summary, the conversation is discussing the best way to solve for the square root of x^2-1 using substitution. Two methods are suggested, one using u = x^2-1 and the other using x = \cosh t. Both methods involve taking the derivative of the inside function and then substituting it into the original equation.
Prototype
(x) sq. root of ((x^2) - 1)?

notice that the derivative of the inside is the same power as the one outside...

hmm, I'm thinking substitution... lol

let u = x^2 - 1
du = 2x dx
dx = du/2x

therefore you will then get (in terms of u)

1/2* (integral of sqrt(u))

$$1/2\int\sqrt{u}du$$

or somethign like that..

hope you can do the rest

PS, is this an ap calc question?

Last edited:
I would use the wonderful substiution:

$$x = \sec (u)$$

Or you could notate like this...

$$u = x^2-1$$

$$du = 2x dx$$

$$\frac{du}{2} = x dx$$

Last edited by a moderator:
Jameson said:
That last step is a typo or it's incorrect.

It should be:

$$u = x^2-1$$

$$du = 2x dx$$

$$\frac{du}{2} = x dx$$

Your end substitution is correct, with $$\frac{1}{2}\int\sqrt{u}du$$
but that one step is off.

can you explain why it is wrong?

i only know through calc bc, got the ap tommorow... if that explains it or not...

it would make sense to let dx = du/2x, so that i could just substitute it in

can't i treat dx and x as separate variables?

I believe Jameson is just worried about your notation. du/(2x)=dx would be okay.

As well, Zurtex' method will work, but it's not necessary as the other suggestion in this thread is easier.

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It's not wrong, either way works:

xdx= du/2 so that $$x\sqrt{x^2-1)$$ becomes u1/2du/2 or

dx= du/2x so that $$x\sqrt{x^2-1}$$ becomes xu1/2du/2x and the x's cancel.

I would suggest the delicious substitution

$$x=\cosh t$$

Daniel.

Ah, sorry, your notation does work. It just takes one extra step of cancelling the x's. I wasn't paying close enough attention. I fixed my post. Sorry for the messup.

Jameson

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