 Let's take another look at an example of an integral involving both tangent and secant as in this example here Let's integrate tangent of x times secant squared of x dx and Well based upon the techniques we've learned so far there's a couple there's actually in multiple ways one could do this So if we try the idea of a u substitution, we might notice. Oh, yeah, there's an even number of secants here So I'm gonna take u to be tangent of x. I'm gonna take du to be secant squared x dx That would then convert this thing to look like the integral of Udu which is wonderful finding its anti-derivative to get u squared over 2 And then plus a constant right and then substituting back in that u equals tangent We end up with one half Tangents squared x plus a constant and so we see that the anti-derivative would be this function right here. Okay That's pretty simple, right? Now what I want to do is actually show you in comparison then, you know, actually Yeah, we have even number of secants, but we also have we could also since we have one tangent we could take one tangent and The secants and we could do a u substitution there, right where we take u to be tangent x Secant sorry, that's the du. We'll take we'll take u to be secant x and And it'll take du to be secant x tangent x du a DX excuse me in that situation Notice that the tangent secant becomes a du and then the other secant becomes a u you end up with u du The exact same integral from before so it's anti-derivative will be the same u squared over two plus a constant But the important difference here is the interpretation of u is different in which case when you write that out You're gonna get one half secant squared x plus constant And so this is the anti-derivative and so then you might look at that for a second like wait a second One half tangent squared versus one half secant squared. How can those both be the anti-derivative? Those are different functions, but did I what one of these steps a mistake? Did I do something wrong and you can you can check the the calculus here? We did everything right those are both legitimate u substitutions or anti-derivatives were perfectly fine How could this be well the thing that the sort of the resolve this concern here? I want to remind you about a very important trigonometric identity We've seen before notice that if I take one plus tangent squared x This is equal to secant squared of x are more importantly if you rewrite it secant squared minus tangent squared is equal to one And so the these two fun because that's what's different between these two anti-derivative You have a tangent squared versus a secant squared So notice by this Pythagorean identity tangent squared and secant squared only differ by a constant Right they differ by a constant and the significance thing in here is this plus C is that constant? Right if you were just to say oh the anti-derivative is one half cos tangent squared And you're like someone else is like but that the anti-derivative is one half secant squared Well, they can't both be right because those are different functions But if you're like oh the anti-derivative is one half secant squared plus a constant or one half tangent squared plus a constant Then since tangent squared and secant squared differ by a constant that plus C makes up the difference And so I like this example because for me when I was a student this example really clarified to me the Importance of this plus C and why did my professor emphasize it so much to me? Why do we need the plus C? It's not just some afterthought it adds a legitimate part of this anti-derivative without it We would be incorrect But with it we can be correct as well as our neighbor who could be correct using a different technique Getting what a seemingly is a different anti-derivative. They're actually equal to each other because of the plus C So don't forget your your good old gelatinous cube right there plus C