yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

_-substitution: defining _ (more examples) | AP Calculus AB | Khan Academy


3m read
·Nov 11, 2024

What we're going to do in this video is get some more practice identifying when to use u-substitution and picking an appropriate u. So, let's say we have the indefinite integral of natural log of X to the 10th power, all of that over X, DX.

Does u-substitution apply, and if so, how would we make that substitution? Well, the key for u-substitution is to see: do I have some function and its derivative? You might immediately recognize that the derivative of natural log of X is equal to 1 over X. To make it a little bit clearer, I could write this as the integral of natural log of X to the 10th power times 1 over X, DX.

Now it's clear we have some function, natural log of X, being raised to the tenth power, but we also have its derivative right over here, 1 over X. So, we could make the substitution; we could say that U is equal to the natural log of X. The reason why I pick natural log of X is because I see something: I see its exact derivative here, or something close to its derivative—in this case, it's its exact derivative.

And so then I could say D u DX is equal to 1 over X, which means that D U is equal to 1 over X DX. And so here you have it; this right over here is d u, and then this right over here is our u. So this nicely simplifies to the integral of U to the 10th power, U to the 10th power D U.

And so you would evaluate what this is, find the antiderivative here, and then you would back substitute the natural log of X for u.

And to actually evaluate this indefinite integral, let's do another one. Let's say that we have the integral of—let's do something interesting here. Let's say the integral of tan(X) DX. Does u-substitution apply here?

And at first, you might say, well, I just have a tangent of X; where is its derivative? But one interesting thing to do is, well, we could rewrite tangent in terms of sine and cosine. So we could write this as the integral of sine of X over cosine of X DX.

And now you might say, well, where does u-substitution apply here? Well, there's a couple of ways to think about it. You could say the derivative of sine of X is cosine of X, but you're now dividing by the derivative as opposed to multiplying by it. But more interestingly, you could say the derivative of cosine of X is negative sine of X.

We don't have a negative sine of X, but we can do a little bit of engineering. We can multiply by negative one twice. So we could say the negative of the negative sine of X, and I stuck one of them. You could say negative one’s outside of the integral, which comes straight from our integration properties. This is equivalent; I can put a negative on the outside and a negative on the inside so that this is the derivative of cosine of X.

And so now this is interesting; in fact, let me rewrite this. This is going to be equal to negative the negative integral of 1 over cosine of X times negative sine of X DX.

Now, does it jump out at you? What you might be? Well, I have a cosine of X in the denominator, and I have its derivative. So what if I made U equal to cosine of X? U is equal to cosine of X, and then D u DX would be equal to negative sine of X. Or I could say that D U is equal to negative sine of X DX.

And just like that, I have my D u here, and this, of course, is my U. And so my whole thing has now simplified to it's equal to the negative indefinite integral of 1 over U, 1 over U D U, which is a much easier integral to evaluate. And then, once you evaluate this, you back substitute cosine of X for U.

More Articles

View All
Momentum collision graphs
A cart of mass m moving rightward at speed 2v hits a slower moving cart of mass m moving rightward at speed v. When the carts collide, they hook together. There’s friction between the track and carts and between the moving parts of the carts. Which of the…
Standard potential, free energy, and the equilibrium constant | AP Chemistry | Khan Academy
For a generic redox reaction, where the reactants turn into the products, the free energy is related to the potential for the redox reaction. The equation that relates free energy and potential is given by: ΔG = -nFE. ΔG is the instantaneous difference …
The Top 5 BEST Investing Apps
What’s up you guys, it’s Graham here. So, as usual, I always read all of the comments, and if you guys ask me to make a specific video, I will go ahead, I will listen, I will make the video. And this video is exactly that, because recently I’ve received n…
Chase a Wild Buffalo Stampede With These Heroic Cowboys | Short Film Showcase
[Music] Big day we’re all been waiting for. Buffalo waiting; all of us are excited, a little nervous, but I guess we’re ready—ready as we’re going to be. My name is Duke Phillips. I manage the Mana Ranch. It’s um, a little over 100,000 acres located in t…
6 NEW Stocks Warren Buffett is Buying!
Warren Buffett runs the most closely followed portfolio in all of investing. Put simply, when Buffett’s company, Berkshire Hathaway, starts buying up stock in a certain company, people pay attention. I frequently say that probably the best way to learn ab…
Checks on the judicial branch | US government and civics | Khan Academy
In other videos, we have talked about how the other branches of government can limit Supreme Court powers. We’re going to continue that conversation in this video by discussing how the amendment process can also limit or overrule a Supreme Court decision.…