 In this video we provide the solution to question number four for practice exam number four for math 1210 In which case we're asked to find the most general Antiderivative of the function r of theta equals cosecant theta times cotangent theta Plus 2 times e to the 2 theta so if we look at this piece by piece since we have a sum of two functions We have to find their antiderivatives individually So let's first look at cosecant theta times cotangent theta now the temptation is to be like Oh the antiderivatives cosecant, but that's not exactly right If you take the derivative of cosecant theta you actually get negative cosecant theta cotangent theta And that's not exactly what we have right here now by antiderivative rules if you take the derivative of negative cosecant theta That'll equal negative negative cosecant cotangent. So a double negative is positive you get cosecant theta Times cotangent theta. So that's exactly what we want This thing is going to pull up to become a negative cosecant theta the issue here is when you take the derivative of secant you get positive secant times positive Tangent so we might conflate the two the antiderivative here is gonna be a negative cosec cosecant of theta. What about the other part? What about this 2? Times e to the 2 theta well the antiderivative there by the usual sort of reverse chain rule in this situation You can use a U substitution I suppose although that's the topic you probably haven't learned by now But this this is one this is an elementary one enough that we could check that you're gonna get that the antiderivative is e to the 2 theta So you put those things together your antiderivative r of theta It's gonna look like negative cosecant theta plus e to the 2 theta plus a constant Don't forget that plus constant because we need the most general antiderivative So that would give us as the correct answer choice f now at the very least If you're if you do struggle with antiderivatives, hopefully you're pretty good at calculating derivatives, right? So what you could do is you could go through each of these options one by one and calculate the derivative and see which happens You know, which one is which is the correct one like if you're like, how did you get the antiderivative to e to the 2 theta? It was e to the 2 theta. I still struggle with that. I'm not very good You could actually take the derivative each of these six options and go from there now You are looking for the most general antiderivative So if there's no plus c that is if the plus c is missing you can throw those out Automatically and you could take the four remaining answers and calculate their derivatives and see which one's the correct So it's kind of working the problem backwards But that is a legitimate strategy for this multiple choice question Although if you are comfortable with computing antiderivatives, it's probably faster just to go with the antiderivative Instead of working all four answers backward