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Evaluate the given integral.$$\int \frac{(\ln x)^{N}}{x} d x$$

$$\frac{(\ln x)^{N+1}}{N+1}+c \text { when } N \neq-1$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 3

The Substitution Method

Integrals

Missouri State University

Campbell University

Harvey Mudd College

University of Nottingham

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.

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the derivative of Ln of X is one over x. So let's take you to be l Innovex. That way we can get rid of one over x DX because now do you? That's just one over x dx and we can rewrite this integral as the integral of you to the end which is equal to you to the end, plus one all over in plus one plus c. And this is assuming that n is not equal to minus one. So if we make that assumption, then we get that this is the integral of you. To the end is U to the n plus one all over end plus one plus e. And then we replace you by Ellen of X. We get that, that's Eleanor vax to the end, plus one all over and plus one plus c. But what happens if an is equal to minus one? Well, in that case, we get that this is the integral off one over X times Ln X, and we can use the same substitution, namely U equals Ellen of x again. Do you? Is one over x dx and we can rewrite this integral in terms of you. So this is the integral of one over you, which is equal to Ellen of the absolute value of you, plus C and you is equal to Ellen of X. So this is Ellen of the absolute value of Ellen of X, plus C. And now weaken. Um, Now we can show that this integral the one that we originally had has two possibilities depending on the value of n. So for one case, we have that this is Allen of X to the end, plus one all over and plus one plus C. And that's if it is not, um, if it is not equal to minus one and then we have that this could be Ellen of the absolute value of Ellen of X plus C. And that's if an is equal to minus one, and that completes the problem.

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