Logarithmic mean - Wikipedia
Derivatives Basic · Differentiation Rules logx, (x ∙ lnx - x) / ln(10) + c. logax, x( logax Example 1: Solve integral of exponential function ∫ex32x3dx. Solution. The integral of many functions are well known, and there are useful rules to work out the integral of more complicated functions Reciprocal, ∫(1/x) dx, ln|x| + C. integrate certain functions to give a natural logarithmic function. It the rules are clear-cut. ln. 1 this integral fits the pattern with () x xf = and () 1. ' = xf.
Let me just copy and paste that. We already established that this is the same thing as two to the x to the third power. Copy and paste, just like that. Then let me close it with a d x. I was able to get it in terms of e as a base. That makes me a little bit more comfortable but it still seems pretty complicated.
- List of integrals of logarithmic functions
- 𝘶-substitution: definite integral of exponential function
- Integration by parts: ∫ln(x)dx
You might be saying, "Okay, look. Well that's going to be three x squared times the natural log of two, or three times the natural log of two times x squared. That's just a constant times x squared. We already have a x squared here so maybe we can engineer this a little bit to have the constant there as well. Let's think about that. If we made this, if we defined this as u, if we said u is equal to x to the third times the natural log of two, what is du going to be?
We could actually just change the order we're multiplying a little bit.
We could say that this is the same thing as x squared times three natural log of two, which is the same thing just using logarithm properties, as x squared times the natural log of two to the third power.
Three natural log of two is the same thing as the natural log of two to the third power. This is equal to x squared times the natural log of eight.
Exponential and Logarithmic Integration – She Loves Math
Let's see, if this is u, where is du? Oh, and of course we can't forget the dx. This is a dx right over here, dx, dx, dx. Where is the du? Well we have a dx. Let me circle things. You have a dx here, you have a dx there. You have an x squared here, you have an x squared here. So really all we need is, all we need here is the natural log of eight. Ideally we would have the natural log of eight right over here, and we could put it there as long as we also, we could multiply by the natural log of eight as long as we also divide by a natural log of eight.
We can do it like right over here, we could divide by natural log of eight. But we know that the anti-derivative of some constant times a function is the same thing as a constant times the anti-derivative of that function.
We could just take that on the outside. It's one over the natural log of eight. Let's write this in terms of u and du. This simplifies to one over the natural log of eight times the anti-derivative of e to the u, e to the u, that's the u, du. This times this times that is du, du.
And this is straightforward, we know what this is going to be. This is going to be equal to, let me just write the one over natural log of eight out here, one over natural log of eight times e to the u, and of course if we're thinking in terms of just anti-derivative there would be some constant out there.
Then we would just reverse the substitution. And we've seen this multiple times. So when you figure out what should be f and what should be g, for f you want to figure out something that it's easy to take the derivative of and it simplifies things, possibly if you're taking the derivative of it. And for g prime of x, you want to find something where it's easy to take the antiderivative of it.
So good candidate for f of x is natural log of x. If you were to take the derivative of it, it's 1 over x.
Let me write this down. So let's say that f of x is equal to the natural log of x. Then f prime of x is equal to 1 over x. And let's set g prime of x is equal to 1.
So g prime of x is equal to 1. That means that g of x could be equal to x.
And so let's go back right over here. So this is going to be equal to f of x times g of x. Well, f of x times g of x is x natural log of x.
So g of x is x, and f of x is the natural log of x, I just like writing the x in front of the natural log of x to avoid ambiguity. So this is x natural log of x minus the antiderivative of f prime of x, which is 1 over x times g of x, which is x, which is xdx. Well, what's this going to be equal to?
Well, what we have inside the integrand, this is just 1 over x times x, which is just equal to 1.