 In the previous video, we utilized the anti-derivative tangent to help us calculate the anti-derivative tangent cubed. What do we do if you just have a secant? Now I wanna point out to you that if you just have a, if you have a secant squared, life is wonderful for you because we know the anti-derivative secant squared is a tangent, tangent x plus c. So that one's pretty simple right there. But if you have just a secant, what do you do? Cause we can't really utilize a Pythagorean identity because we need at least a secant squared. Otherwise, square roots get into the game and that's not gonna help us all. We can't do the other substitute substitution because we need a tangent and a secant to make that one work. So this one might have you scratching your head for quite a while. This one's a bit crafty. You have to be very crafty on this one. But as we're gonna use other examples where we need to know this anti-derivative, this is one sort of just telling you sort of like an ivory tower type problem. One might not have expected how in the world one ever came up with this, but this is one you're gonna wanna memorize and this is the derivation of it. So what we're gonna do is very cleverly multiply by a strategic number one. We're gonna multiply the top by secant tangent, a secant x plus tangent x. And then we have to do the same thing to the bottom, secant x plus tangent x. So we're gonna times the top and bottom by that and it's like, well, why would we do that? Well, we're gonna see in a moment why this is fruitful. Distribute the secant throughout the numerator. And when you do that, you're gonna have a, well now the numerator becomes secant squared x plus secant x tangent x. And this sits above secant x plus tangent x. And don't forget your dx here. So this is a really clever multiplication because you'll now notice in the numerator we have a secant squared. A secant squared is a derivative, right? A secant squared is the derivative of tangent. Oh, looky here. There's our good friend right there. And then on the other hand, if you look at secant x tangent x, it's likewise a derivative and it's the derivative of, oh look, secant x. So with this multiplication we did here, we actually set up for a very nice U substitution. Set U to be secant x plus tangent x. Don't know why there's a D there. Secant x plus tangent x, then du would equal, the derivative of secant is secant x tangent x. The derivative of tangent is secant squared x dx. And that's exactly the situation we have right here. This integral will then become the integral of du over u. How fantastic. That's really great. And so the anti-derivative of du over u, of course, is the natural log of u. So we get the natural log of the absolute value of u plus a constant. And then replacing u with secant plus tangent, we see the anti-derivative will be the natural log of the absolute value of secant x plus tangent x plus a constant, like so. And so it's really nice trick that basically works only for this situation. This is not some technique we can sort of expand on a mass scale. This helps us find the anti-derivative of secant. A modification of this could be used to find the anti-derivative of cosecant, of course. But beyond that, this technique, it was perfect for this situation. Using the fact there's a similarity between the derivatives of tangent and secant. That's kind of what we played off with this right here. And so this anti-derivative is one you're gonna wanna memorize. You can memorize the technique here or you might just wanna memorize this for future reference. We'll see six examples where we do need to know explicitly what is the anti-derivative of secant. And that'll be coming very soon.