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The Integral of Csc(θ)

What is the integral of cscθ? Usually this is done by telling you to multiply by blah/blah where blah is some expression that comes seemingly from out of nowhere. That's a fun trick, but I want to show you how you can figure out what that expression should be with a little bit of playing around!

This is a good lesson in and of itself - if you're stuck and don't know where to go, just play around with it and see if your eye catches any patterns!

Since we don't know the integral, let's take the derivative of cscθ.

Notice the derivative of cscθ has cscθ as a factor. The other factor is cotθ, so let's investigate that by taking its derivative as well.

So the derivative of cotθ involves cscθ. It's like a little loop - and that's what we can take advantage of. Something magical happens if we add these derivatives together:

Dividing both sides by cscθ + cotθ gives

which should scream to us: LOGARITHMIC DERIVATIVE! The equation becomes

which allows us to finally say

Isn't that fun? (P.S. Don't forget your +C )