# This is a crazy integral!

1. May 25, 2012

### tempneff

1. The problem statement, all variables and given/known data

$$\int \sqrt{a-\frac{1}{x}}dx$$

2. Relevant equations

U-sub
Trig Sub

3. The attempt at a solution
I checked the integral with values: a=2, from x=1 to x=2.....but it gave a different value then I get with mine. They did it a far different way to, but I can't see why this won't work....can you?

Evaluate:$$\int \sqrt{a-\frac{1}{x}}dx$$
$$\int \sqrt{a-\frac{1}{x}}dx \rightarrow \int \sqrt{\frac{ax-1}{x}}dx\rightarrow \int \frac{\sqrt{ax-1}}{\sqrt{x}}\rightarrow \int \sqrt{\frac{1}{x}}\sqrt{ax-1}$$
Let $$u=\sqrt{ax-1} \hspace{15pt}u^2=ax-1\hspace{15pt}x=\frac{u^2+1}{a}\hspace{15pt}\sqrt{ \frac{1}{x}}=\sqrt { \frac{a}{u^2+1}}$$
and $$du=\frac{a}{2\sqrt{ax-1}} dx\rightarrow dx=\frac{2\sqrt{ax-1}}{a} du \rightarrow \frac{2u}{a}du$$
Now
$$\int \sqrt{a-\frac{1}{x}}dx\rightarrow \int \frac{2\sqrt{a}}{a} \frac{u^2}{\sqrt{u^2+1}} du\rightarrow \frac{2}{\sqrt{a}}\int \frac{u^2}{\sqrt{u^2+1}}du$$
Let $$u=\tan \theta \hspace{15pt}\sqrt{u^2+1}= \sec \theta$$
and
$$du =\sec^2 \theta d \theta$$
so
$$\frac{2}{\sqrt{a}}\int \frac{u^2}{\sqrt{u^2+1}}du \rightarrow \frac{2}{\sqrt{a}}\int \frac{\tan^2 \theta}{\sqrt{\tan^2 \theta +1}}\sec^2 \theta d \theta \rightarrow \frac{2}{\sqrt{a}}\int \frac{\tan^2 \theta}{\sqrt{\tan^2 \theta +1}}\sec^2 \theta d \theta$$
$$\frac{2}{\sqrt{a}}\int \frac{\sec^2 \theta +1}{\sqrt{\sec^2 \theta}}\sec^2 \theta \rightarrow \frac{2}{\sqrt{a}}\int \sec^3 \theta + \sec \theta d \theta$$
By Integral Table $\# 82$ and $\#84$
$$=\frac{2}{\sqrt{a}}\left( \frac{1}{2}\sec \theta \tan \theta +\frac{3 }{2} \ln |\sec \theta + \tan \theta |+C \right)$$
where $$u=\tan \theta \hspace{15pt}\sqrt{u^2+1}= \sec \theta$$
so $$\frac{2}{\sqrt{a}} \left( \frac{1}{2}u\sqrt{u^2+1}+\frac{3}{2}\ln | \sqrt{u^2+1}+u|+C \right)$$
where
$$u=\sqrt{ax-1}$$
so
$$\frac{2}{\sqrt{a}}\left( \frac{1}{2}\sqrt{ax-1}\sqrt{ax}+\frac{3}{2} \ln |\sqrt{ax}+\sqrt{ax-1}|+C\right)$$
or
$$\sqrt{ax^2-x}+\frac{3}{\sqrt{a}} \ln |\sqrt{ax}+\sqrt{ax-1}|+C$$

2. May 25, 2012

### Jorriss

I take back my comment. It may not work. You could try u = a - 1/x but I dont know if it works out fully. This isnt integral I thought it was at first.

Last edited: May 26, 2012
3. May 26, 2012

### tempneff

Yeah I saw it completed with $$u=\frac{ax-1}{x}$$ and I will go that way if I have to, but I really want to know why this isn't working. I broke a rule somewhere....

4. May 26, 2012

### tt2348

maybe try x=(sec(theta)^2)/a, that might clean up some of it

5. May 26, 2012

### andrien

check out,you have put tan^2(theta)=sec^2(theta)+1, it will be sec^2(theta)-1.just above where you have written 'by integral table'.

6. May 26, 2012

### tempneff

Excellent that did it!!!!!!

Last edited: May 26, 2012