1. ## Integration by substitution

Hi

I have the following integral but I cant see how to solve it, I know it involves substitution.

$\displaystyle \int\dfrac{1}{\sqrt{x^2-4}}$

Any hints would be appreciated

James

2. Try $\displaystyle x=2sec(\theta)$
Do not forget the $\displaystyle dx$ in your integral.

3. Hi

I tried using it but didn't get very far, any chance you give me any further hints. The substitution I've done so far hasn't used trig but the standard $\displaystyle u$.

Thanks

4. When you substitute $\displaystyle x=2sec\theta$, you will face ( Assuming $\displaystyle tan\theta > 0$ ):
$\displaystyle \int \frac{2 sec\theta tan\theta}{2 tan\theta} d\theta$

You can not solve it?

5. Substituting $\displaystyle u=\sqrt{x^2 - 4}$ will give another integral requires a trigonometric substitution.

6. Hi

In our handbook there is a list of standard integrals, this one gives

$\displaystyle \int\dfrac{1}{\sqrt{x^2-a^2}}=\ln(x+\sqrt{x^2-a^2})$

I could use this but they obviously want to see workings, any ideas how they got this?

James

7. By the same substitution $\displaystyle x=asec\theta$ .

8. Thanks, got it, sorry it took a while.