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

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In summary, to integrate 1/(x^2-1)^.5, make a u-substitution with u=x^2-1 and du=2x. For x/(x^2-1)^.5, make a trigonometric substitution with x=secø and dx=secøtanødø. The integral will then become ln|x+√(x^2-1)| + constant.
  • #1
ComFlu945
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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
 
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  • #2
For the first, make a u-substitution.

For the second, make a trigonometric substitution suggested by that difference of squares.
 
  • #3
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
Let x=secø
dx=secøtanødø
tanø=√(x2-1)

So your integral becomes:
∫dx/√(x2-1) = ∫secødø = ln|secø+tanø|

Substituting back in
ln|secø+tanø|=ln|x+√(x2-1)|
 
  • #5
ComFlu945 said:
integral ( sinh(y)/sinh(y)) dy = 1 + constant

Think about that some more
 
  • #6
By u substitution, let u=x^2-1 then du=2x
you have x you only need 2
 

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

To integrate x/(x^2-1)^.5, you can use the substitution method by letting u = x^2-1. This will give you the integral of 1/u^0.5, which can be solved using the power rule.

2. Is there another way to integrate x/(x^2-1)^.5?

Yes, you can also use the trigonometric substitution method by letting x = secθ or x = tanθ. This will give you the integral of secθ or sec^2θ, respectively, which can be solved using trigonometric identities.

3. What is the result of the integral of x/(x^2-1)^.5?

The result of the integral of x/(x^2-1)^.5 is ln|x+(x^2-1)^0.5| + C, where C is a constant of integration.

4. Can I use integration by parts to integrate x/(x^2-1)^.5?

No, integration by parts is not a suitable method for solving this integral. It is better to use substitution or trigonometric substitution.

5. Can I simplify the integral of x/(x^2-1)^.5?

Yes, you can simplify the integral by using algebraic manipulation to rewrite it as (x^2-1)^-0.5, which can then be integrated using the power rule.

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