 Let's look at another example of an integral. This time, let's find the antiderivative of the function x over the square root of x squared plus four. Now in this example, you can see that this square root of x squared plus four, that being in the denominator really indicates to me that, hey, we could use a trigonometric substitution to help us out here, right? Because after all, if we use the substitution, the second type, that is the tangent substitution, we could set x equals to two tangent and we could simplify, or we could convert the integral into trigonometric form from there and then we could calculate the antiderivative. Not so bad, right? But what they really wanna mention is that although trigonometric substitution is a very valuable tool and it would work successfully in this situation, it might not be the best tool, right? You could take x to be two tangent and that would be very fruitful, but I actually do have a other recommendation in this one. Instead, if we take u to be x squared plus four, notice that du would then equal two x dx for which if we times by two and times by one half there, we can do a u substitution where this thing would become one half the integral of du over the square root of u or more simply one half u to the negative one half power du. So then by the power rule, we end up with one half, well, u, you raise the power to be one half, you'll then divide by one half for which case we should just mention that the one half just cancel each other out entirely plus a constant and then replacing u with the original variable x, we're gonna get the square root of x squared plus four plus an arbitrary constant. And so you can see that in this situation, the u substitution was a much simpler approach than trigonometric substitution. So this one can be very challenging about integrals at times is that we have so many different techniques, we have the u substitution, the trigonometric substitution, just as two variants of the substitution method, we didn't even touch integration by parts here. It can be challenging at times to figure out what is the right technique to use. And this kind of comes with experience and practice. You gain an intuition of things after a while. Now, by all means, if you try to compute this with trigonometric substitution, that wouldn't be bad, it just would be a little bit longer, right? And so before you necessarily jump into the pool on trig subs or integration parts or whatever, I would recommend that you try to consider what are alternative strategies I could use and then pick the path that's gonna be the most optimal, it's the fastest way to compute it. So don't just automatically assume trigonometric substitution, might wanna keep in mind, well, what a standard u substitution work in this situation, which we saw right here. And so that brings us to the end of our lecture, number 12, about trigonometric substitutions. I do wanna continue this discussion in the next lecture 13, so check out that video to learn some more about trig subs, especially for like a secant substitution, which we haven't done yet. But that all officially in lecture 12 right here. So thank you for watching. If you have any questions on any of these lecture videos, please click and post them in the comments below. I'll be happy to answer them. Like this video, subscribe to it if you wanna see some more videos like this in the future. And I will see you next time, bye everyone.