1. ## integral with radical and trigonometry

Is there anyone who could walk me through this practice problem?:

$\int{\sqrt{1 + \frac{cosx}{sinx}}}\,dx$

I really struggle with integrals involving radicals or trigonometry.

2. ## Re: integral with radical and trigonometry

Hey infraRed.

There is probably a simpler way but one suggestion I have is to use complex numbers. cos(x) = [e^(ix) + e^(-ix)]/2 and sin(x) = [e^(ix) - e^(-ix)]/2i.

Given that you probably haven't done a lot of stuff like that can you mention the identities and concepts that you have studied in the course that you are taking?

3. ## Re: integral with radical and trigonometry

My apologies, but I actually misread the problem. (cos(x)/sin(x)) is supposed to be (cos(x)/sin(x))^2. I think that makes the problem a lot more workable.

4. ## Re: integral with radical and trigonometry

Hello, infraRed!

$\displaystyle \int \sqrt{1 + \left(\frac{\cos x}{\sin x}\right)^2}\,dx$

Apply some trig identities . . .

$\sqrt{1 + \left(\frac{\cos x}{\sin x}\right)^2} \;=\; \sqrt{1 + \cot^2x} \;=\;\sqrt{\csc^2x} \;=\;\csc x$

So we have: . $\displaystyle \int \csc x\,dx$

Got it?

5. ## Re: integral with radical and trigonometry

Originally Posted by Soroban
Hello, infraRed!

Apply some trig identities . . .

$\sqrt{1 + \left(\frac{\cos x}{\sin x}\right)^2} \;=\; \sqrt{1 + \cot^2x} \;=\;\sqrt{\csc^2x} \;=\;\csc x$

So we have: . $\displaystyle \int \csc x\,dx$

Got it?
Which of course then can be written as

\displaystyle \begin{align*} \int{\csc{(x)}\,dx} &= \int{\frac{1}{\sin{(x)}}\,dx} \\ &= \int{ \frac{\sin{(x)}}{\sin^2{(x)}}\,dx} \\ &= -\int{\frac{-\sin{(x)}}{1 - \cos^2{(x)}}\,dx} \\ &= -\int{\frac{1}{1 - u^2}\,du} \textrm{ after making the substitution } u = \cos{(x)} \implies du = -\sin{(x)}\,dx \end{align*}

I'm sure you can go from here