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(a) Compute $\frac{d}{d x}(x \ln x-x)$. (b) What function can you now antidifferentiate?

(a) $\ln x$(b) $\ln x$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 1

Anti differentiation - Integration

Integrals

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Lectures

05:53

In mathematics, an indefinite integral is an integral whose integrand is not known in terms of elementary functions. An indefinite integral is usually encountered when integrating functions that are not elementary functions themselves.

40:35

In mathematics, integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function of a real variable (often called "the integrand"), an antiderivative is a function whose derivative is the given function. The area under a real-valued function of a real variable is the integral of the function, provided it is defined on a closed interval around a given point. It is a basic result of calculus that an antiderivative always exists, and is equal to the original function evaluated at the upper limit of integration.

02:15

(a) Compute $\frac{d}{d x}…

02:27

(a) Explain why you can re…

01:14

Calculate.$$\int \frac…

02:07

Find $\int \frac{d x}{(x \…

01:36

Find $\int x \ln x \mathrm…

00:23

$\int \ln 2 x d x=$(A)…

01:02

00:33

$\int \frac{\ln x}{3 x} d …

the thought process in this problem is and not all right then. Is there asking you to find the derivative of X natural log of X minus X and I would use the product rule on this. So if you do the product tool on the left side while the derivative of excess one, you leave natural with excellent. Um, Plus, now you leave X alone times the derivative of natural log of X, which is one over X, and then you have to take the derivative of X, which is one. As you notice in this problem, these exes will eliminate S O you're left with. I just write out one times Anything is that thing natural log of X plus on minus one. We'll look at that. These these ones canceled. So we're looking at the final answer. That is just natural Log of X. So there's your answer. Part A. So what that means is, if the derivative of this piece is natural, log of X, then what that means is that the integral of natural log of X dx is now equal to that thing. We just did. Ex natural log of X minus X um plus C. I don't know if they want you to do this whole thing. Um, you know. But now we can find the anti derivative of natural log by doing this formula. And don't forget about. Plus, C is a directive of a constant zero. So we need that. There you go. So

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