 Alright, I want to calculate another antiderivative. Let's find the antiderivative e to the square root of x This is another one that illustrates this technique of the rationalizing substitution is this is a pretty friendly one here Right, you're gonna take u equal to the square root of x and then du will equal dx over 2 square root of x and since the square root of x is u This is the same thing as saying 2u du equals dx That is we took this equation right here We cleared the denominators and replaced the square root of x with u And so if you make those substitutions in there We then get the integral of e to the u is the first part and then dx becomes 2u du Like so or we could write this as 2 integral of u e to the u du And this is a classic one to do integration by parts with For which if we're doing integration by parts, we'll set u equal to u. That's pretty convenient And then dv will equal e to the u du u And that way du equals du that was also convenient and then v equals e to the u like so And then applying that to this function right here. We get two times u e to the u minus the integral of e to the u du For which we can see very quickly that the antiderivative e to the u is itself e to the u plus a constant So our antiderivative respect to u is 2u e to the u Plus 2 e to the u plus a constant and then replacing Replacing this u with the square root of x we get 2 square root of x e to the square root of x Plus 2 e to the square root of x plus a constant for which you could you could factor that thing There's a 2 and a e to the square root of x that's common to both of them And so this u or this rationalizing substitution works out really nicely a lot of the times you get something like the following Oh, whoops. I made a I made a mistake. Let me fix that There should be a minus sign here A minus sign was a consequence of the integration by parts Like though, sorry about that mistake. So rationalizing substitution works out really nicely right because the idea is The square root is self similar to its derivative and you can manipulate that similarity to help you with these substitutions