Raw Transcript
Hello, everyone. Tom from everystepcalculus.com and everystepphysics.com. I’m going to do a U Substitution problem. Basically a simple one but nothing is simple in Calculus unless you know what you’re doing. It certainly helps to have a photographic memory. This problem was sent to me by a student. I’m going to do it on the video and show him how to do it and show you how to do it. Index 8 to get to my menu. U Substitution. You’d scroll down to it there. Choose that. We’re going to enter the function. Alpha, left parentheses 6 times x minus 5, close off the parentheses to the 4th power. So here’s what we have. I say it’s okay. So let’s talk a little bit about U Substitution. You should do derivatives in your head within one second. What’s the derivative of this function in here? It’s 6. There’s no x’s nothing, it’s just 6. And you have dx on the outside. Okay. So there’s no x or nothing else on the
outside except dx. So let’s, that’s what you choose for U. So U is 6x minus 5, derivative is 6dx, anytime you see a constant on here. This is the trick that always screwed me up in school. You need to take this out of this side and isolate the dx. Okay, because remember the dx is by itself on the function and now this dx needs to be by itself like here. So you divide both sides by 6 through Algebra. DU divided by 6 equals dx. It variably happens like this. Don’t think it doesn’t. Almost every problem you do on U Substitution, your’e going to take the constant from this side and put it over on this side. Okay. So no U equals 6x minus 5. Therefore, the integral of u to the 4th is the function, u to the 4th times du divided by 6. Now here’s the du we just did, du divided by 6. In integration, the constants always have to come out of the integration. So you have to take one sixth out which I do here. Here’s one sixth out of the integral. And here we have the u to 4 by itself, and the du is by itself. Okay. So now we got one sixth times, and now we do the integral,u to the 5 divided by 5. Okay. So now this constant has to come out of here, too. So we have one sixth times one fifth times u to the 5 plus C. Well one sixth times one fifth is one 30th times 6 and then we substitute back in for u, 6 times x minus 5 to the 5 power plus C. Every U substitution is not in the system and if you learn this system, you’re going to be alive but the main thing is to be able to do derivatives in your head in one second. In your sleep. The derivative of this right here is 6 and you need to look and see if you can match something on the outside, etc. For U substitution. It’s all tricky though until you do it ten thousand times. Calculus is a language and most the cases when I was going to school, it was learning Latin or learning Spanish. I didn’t know what they were talking about. So, have a good one.
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