# How do I integrate this? x/(x^2-1)^.5

1. Nov 16, 2009

### ComFlu945

How do I integrate this? 1/(x^2-1)^.5

How do I integrate this? x/(x^2-1)^.5

And this
1/(x^2-1)^.5

Last edited: Nov 16, 2009
2. Nov 16, 2009

### g_edgar

For the first, make a u-substitution.

For the second, make a trigonometric substitution suggested by that difference of squares.

3. Nov 16, 2009

### ComFlu945

Thanks!

First one worked like a charm.

For second one I substituted x for cosh(y). Since cosh(y)^2-1=sinh(y)^2, but bottom turns into sinh(y). And since x=cosh(y), dx/dy= sinh(y).

Back to original equation:
integral( 1/(x^2-1)^.5 dx) = integral ( sinh(y)/sinh(y)) dy = 1 + constant. However, the answer is supposed to be cosh^-1(x).

4. Nov 16, 2009

### Matthollyw00d

Let x=secø
dx=secøtanødø
tanø=√(x2-1)

∫dx/√(x2-1) = ∫secødø = ln|secø+tanø|

Substituting back in
ln|secø+tanø|=ln|x+√(x2-1)|

5. Nov 17, 2009