# Integral of $\int \frac{dx}{\sqrt{x^2 -9}}$

$$\int \frac{dx}{\sqrt{x^2 -9}}$$

$x = 3 \sec \theta \implies dx = 3 \sec\theta \tan\theta d\theta$

\begin{align} \int \frac{dx}{\sqrt{x^2 -9}} & = \frac{1}{3}\int \frac{3 \sec\theta \tan\theta d\theta}{\tan\theta} \\ \\ & = \int \sec\theta d\theta \\ \\ & = \ln | \sec\theta + \tan\theta| + c\end{align}

$x = 3\sec \theta \implies \sec\theta = \frac {x}{3}$

$\tan\theta = \frac{\sqrt{x^2 - 9}}{x}$

I have have confused x with 3 but I cannot get the proper answer which is

$$\ln | x + \sqrt{x^2 - 9}| + c$$

I always get $\dfrac{x}{3}$ or $\dfrac{\sqrt{x^2 - 9}}{x}$ or some variation of that, I can't eliminate them to get their answer.

## 2 Answers

What you should end with is

$\sec\theta = \dfrac x3\quad$ and $\quad\tan \theta = \dfrac{\sqrt{ x^2 - 9}}{3}$.

Then you have \begin{align} \log \Big| \frac 13\left(x + \sqrt{x^2 - 9}\right)\Big| + C & = \log|x +\sqrt{x^2 - 9}| -\log 3 + C \\ \\ & = \log|x + \sqrt{x^2 - 9}| + C'\end{align}

• My book doesn't have the 1/3, I have spent about an hour on this proble and I just can't figure out how they got it. – Sailor Jim Jul 4 '13 at 15:38
• The factor $\frac 13$ in the log becomes a constant: $$\log\left(\frac ab\right) = \log a - \log b$$ – Namaste Jul 4 '13 at 15:39
• That is a dirty trick. – Sailor Jim Jul 4 '13 at 15:43
• I agree, you were simply off by a constant of integration. Keep a look out, this "dirty trick" will come up again and again! ;-) – Namaste Jul 4 '13 at 15:45
• @SailorJim Be prepared, it is a constant trick in these matters ;-). (+1, amWhy). – Julien Jul 4 '13 at 15:46

$\theta=arc\sec\frac{x}{3}\implies \sec\theta=\frac{x}{3}$ and then $\tan\theta=\frac{\sqrt{x^2-9}}{3}$ so, integral would be $\ln\left(\frac{x}{3}+\frac{\sqrt{x^2-9}}{3}\right)+c=\ln(x+\sqrt{x^2-9})+k(=-\ln 3+c)$

• How does the inner ln term cancel out their denominators? – Sailor Jim Jul 4 '13 at 15:33
• i made some mistake there. See the updated answer. – Aang Jul 4 '13 at 15:38
• What is k? That part confused me. – Sailor Jim Jul 4 '13 at 15:38
• For indefinite integral constants doesn't matter much , you just collect all the contants and call it another final constant. So first constant of integration $c$ with another constant $-\ln 3$ is just another constant $k$. – Aang Jul 4 '13 at 15:41
• Oh okay I see now. – Sailor Jim Jul 4 '13 at 15:44