How to Use Trig Substitution for Integrals Involving (x²-a²)

[maff is tuff]

Homework Statement

∫ dx/(x² -a² )

Homework Equations

When (x² -a² ) appears in an integrand, you can use the trig sub: x=asecθ right?

The Attempt at a Solution

I know I could solve this using partial fractions but why doesn't trip sub work here? Or does it? I have attached my attempt below. Ignore the bottom half of the page. Thanks all for your help. :)

 

[jhae2.718]

Trig sub works. $$\int \csc\thetad\theta = -\ln|\csc\theta+\cot\theta|$$

 

[maff is tuff]

How would I derive that? I would need to show how to get there to get any credit. Or what would be my first step in deriving that because I have no idea. And thanks for replying to one of my questions again; I recognize you from a couple nights ago.

 

[Voivode]

Change csc to (csc^2 + csccot)/(csc + cot). Then use u-substitution.

 

[SammyS]

A common way to do this integration is to expand 1/(x² - a²) using partial fractions.

(x² - a²) = (x - a)(x + a), therefore:

$$\frac{1}{x^2-a^2}=\frac{B}{x-a}\,+\,\frac{C}{x+a}$$

Multiply both sides by (x - a)(x + a). Find B & C.

Your integral then becomes: $$\int\,\left(\frac{B}{x-a}\,+\,\frac{C}{x+a}\right)\,dx$$

 

[maff is tuff]

Thank you all for your replies. I will try your suggestions.