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Find the general indefinite integral.
$ \displaystyle \int \biggl( x^2 + 1 + \frac{1}{x^2 + 1} \biggr)\, dx $
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00:44
Frank Lin
00:39
Amrita Bhasin
Calculus 1 / AB
Chapter 5
Integrals
Section 4
Indefinite Integrals and the Net Change Theorem
Integration
Harvey Mudd College
University of Michigan - Ann Arbor
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.
00:45
Find the general indefinit…
00:56
Evaluate the indefinite in…
00:51
Find the indefinite integr…
06:20
Evaluate the integral.
02:24
00:47
01:04
All right. There's two parts to this problem that you need to know and that's uh one part is that the integral affects to the end power is equal to adding one to the experiment. Then multiply by the reciprocal of that new experiment. Don't forget about your plus E. And the other piece to know. Is that the integral of one over X squared plus one dx is equal to inverse tangent or marketing? You might be familiar with that of X plus C. So using these two rules and our properties of integral roles, we can find this answer X squared plus one plus that one over X squared plus one D. X. And uh it's pretty straightforward. I'm just going to repeat what I said. You add one to your experiment and multiply by the reciprocal of your experiment. Um technically you could think of this is actually zero power to realize that that's X to the first. Um and just thinking about the derivative of X is one. So that is correct. Mhm. Plus this comes straight from that rule, inverse tangent of X. And then don't forget about Plus C. And if you're saying they're still confused about plus C, it's because we are taking the derivative of our answer to get backwards and the derivative of a constant is zero, and we're not going to write plus zero in here. So we don't know if there's a constant right there enough.
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