# integral with radical and trigonometry

• Sep 27th 2013, 10:06 AM
infraRed
Is there anyone who could walk me through this practice problem?:

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

I really struggle with integrals involving radicals or trigonometry.
• Sep 27th 2013, 08:51 PM
chiro
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?
• Sep 28th 2013, 07:21 AM
infraRed
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.
• Sep 28th 2013, 07:33 AM
Soroban
Re: integral with radical and trigonometry
Hello, infraRed!

Quote:

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

Apply some trig identities . . .

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

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

Got it?
• Sep 28th 2013, 07:39 AM
Prove It
Re: integral with radical and trigonometry
Quote:

Originally Posted by Soroban
Hello, infraRed!

Apply some trig identities . . .

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

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

Got it?

Which of course then can be written as

\displaystyle \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 :)
• Sep 28th 2013, 08:38 PM
ibdutt
Re: integral with radical and trigonometry