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Evaluate the given integral and check your answer.$$\int \frac{4}{x} d x$$

$$4 \ln |x|+c$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 1

Anti differentiation - Integration

Integrals

Harvey Mudd College

Baylor University

Idaho State University

Boston College

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.

01:33

Evaluate the given integra…

01:42

01:17

Evaluate the integrals.

01:32

01:06

Find the indefinite integr…

01:26

02:43

Evaluate the integral.…

02:10

Okay, so there's a quick way of doing this problem. Um, and I'll try my best. Thio help you understand The quick way of doing this is understanding for students that the derivative of natural log of X is equal to one over X. Uh, the only thing is we can Onley log positive numbers and it's not assumed, uh, that this X is positive Eso That's why when we work backwards, we have to state that excess positive. So, like, really, what I should write down is the derivative of natural log of the absolute value of X is equal to one over X. So this would be the fast way of doing this. Um, another rule of integral XYZ. That four is just a constant that you could bring in front so you might rewrite the problem like this. And then you can jump to the final answer as four natural log of the absolute value of X plus c. You have to have that plus e there. Now, sometimes when I teach us to my students, I get them to think of the problem. This was so this is this is, uh might help you is if you think of the problem Has four x to the negative first D X and you try adding one to the exponents. Well, we have an issue because the next step is to divide by zero. Well, you cannot divide by zero. So therefore, this is incorrect. Eso There must be a different way to find this anti derivative. And that different way is through the use of natural law. Um, if you take the derivative of this to check your work, you will get that answer. So we're good. We have the right answer.

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