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Evaluate the given integral and check your answer.$$\int \frac{2}{x} d x$$

$$2 \ln |x|+c$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 1

Anti differentiation - Integration

Integrals

Missouri State University

Oregon State University

Baylor University

Idaho State University

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.

01:50

Evaluate the given integra…

01:43

Evaluate the integral and …

01:32

02:34

The idea of this problem is remembering the derivative. There's a certain derivative that will give me one over X, uh, needs room there one over X, and that derivative is the natural law of X. Now, if we do the derivative of natural log of X, we can Onley log positive numbers, and it's not assume that excess positive in this problem. So that's why we do the absolute value when we work backwards. The other thing I like to point out for my students is that, too is just a constant. So they rule with intervals as we could write that, too, in front, so to integral so two times the angle of the times of one over x DX. And if you follow this rule that it makes perfect sense that we get to natural log of the absolute value of X. And don't forget about your plus seats. This is your correct answer, and you could double check. That's right by doing the derivative, and you'll get this same answer all the way back here to check your work. Now, sometimes I do explain to my students because I always have at least one student well, look at that and they'll try to rewrite this as two x to the negative first power and try to do the integral of that. And it's a good thought process, except if you try to add 12 Now you want to get zero and then you have to divide by zero. Well, that is incorrect. We cannot divide by zero. So there has to be a different function that gives us this anti derivative, which is natural log and were correct.

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