1. ## Trig Substitution

I'm trying to integrate $\displaystyle \int \frac{\sqrt{9-x^2}}{x} \,\, dx$

I let $\displaystyle x = 3\sin \theta \Rightarrow dx = 3\cos \theta \,\, d\theta$

made the substitution $\displaystyle \int \frac{3\cos \theta\sqrt{9(1-\sin^2 \theta)}}{3\sin \theta} \,\, d\theta$

and simplified to $\displaystyle 3\int \frac{\cos^2 \theta}{\sin \theta} \,\, d\theta$ which is where I'm stuck.

I tried to simplify the trig functions into $\displaystyle \cot \theta \cos \theta$ or $\displaystyle \cos^2\theta \csc \theta$ but nothing helped

2. ## Re: Trig Substitution

You can simplify like this:

$\displaystyle \int\frac{1-sin^2(\theta)}{sin(\theta)}d\theta=\int[{csc(\theta)-sin(\theta)] d\theta$

3. ## Re: Trig Substitution

Try a hyperbolic substitution x=3th(theta), it will work.

4. ## Re: Trig Substitution

The substitution

$\displaystyle u^{2}=9-x^{2}$

works reasonably well.

5. ## Re: Trig Substitution

There are a number of ways to do it, but the way ReneG attempted it works fine as well, it was just the remaining trig that had to be worked out.

6. ## Re: Trig Substitution

To: atkinsjr
My calculus 1 class did not involve integrating csc(x), that is why I offered the simpler hyperbolic substitution.