integrate 2t x csc^2t dt

Question

anonymous · Answer

$$\int\limits_{}^{} 2\Theta \csc ^{2}\Theta $$

zepdrix · Answer

`Integration by Parts` \:)/
Do you remember the anti-derivative of csc^2x or maybe -csc^2x?

anonymous · Answer

Looking in my notes

anonymous · Answer

-cotx

anonymous · Answer

ohhhh, so this is a straightforward integration by parts problem?

anonymous · Answer

no crazy trickery?

zepdrix · Answer

Yah it doesn't look like there is anything sneaky going on here :)

zepdrix · Answer

Oh I guess we'll have to deal with the integral of cotx, which you may or may not have in your notes.

That is rather tricky to get through if it's not in your notes :D

zepdrix · Answer

Oh wait wait, I was confusing $\large \int \csc\theta$ with $\large \cot\theta$.
Cotangent shouldn't be too bad actually. :D
Convert to sines and cosines and do a u-sub.

Lemme know if you get stuck anywhere.

anonymous · Answer

umm, I got $$-2Thetacot \Theta+\ln \left| sinx \right|$$

zepdrix · Answer

Only one minor problem.
Your $\large u=2\theta$, so your $\large du=2$
So your integral that involved cot theta should have a 2 in front of it.

And also try to keep everything the same variable :) lol
I see theta's and x's in your solution.
I make that mistake sometimes myself.

anonymous · Answer

I still have the answer wrong :(

zepdrix · Answer

$$\Large -2\theta \cot \theta+\color{royalblue}{2}\ln|\sin \theta|+C$$
Understand where the blue 2 is coming from? :o

anonymous · Answer

wait, no, I got it.  Cool.  Thank you!

anonymous · Answer

I understand what I was doing wrong.