# The Integral of Csc(θ)

What is the integral of ** cscθ**? Usually this is done by telling you to multiply by

**where**

*blah/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!**

*blah*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

**as a factor. The other factor is**

*cscθ***, so let's investigate that by taking its derivative as well.**

*cotθ*So the derivative of ** cotθ** involves

**. It's like a little loop - and that's what we can take advantage of. Something magical happens if we add these derivatives together:**

*cscθ*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 **)

**Thank you for reading!**

**Jonathan Gerhard**

**J****Math****G**

**J**

**Math**

**G**