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Find the general indefinite integral.

$ \displaystyle \int \frac{1 + \sqrt{x} + x}{x} \, dx $

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00:48

Frank Lin

00:32

Amrita Bhasin

02:08

Gregory Higby

Calculus 1 / AB

Chapter 5

Integrals

Section 4

Indefinite Integrals and the Net Change Theorem

Integration

Missouri State University

Campbell University

Baylor University

University of Michigan - Ann Arbor

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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Find the general indefinit…

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Find the indefinite integr…

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Find the Integral of \int …

Alright, so here we have an integral. Looks a little bit messy but we can do a trick and divide each term by X. Um basically we have a common denominator. So we'll just split apart each part and have it divided by X. And then see what we get it cleans up a bit enough that we can solve this. So the first term is one of her acts. The second term I can rewrite as X to the 1/2. Uh Well I'll go ahead and show you the steps extra one half over X. And then we have just one here and so we'll just clean up before we do the anti derivative. We'll go ahead and clean up this middle term. I can subtract exponents. So one half minus one is X to the minus a half. Alright, now we're ready to take the anti derivative. So um the anti derivative of one over X is natural log of absolute value of acts. We're gonna do a reverse power rule on the middle term. So I'm going to add one to the exponents minus a half plus one is positive a half. And we divide by that number, we just put up there and then plus X plus C. So we're almost there. We can and it's just um the middle term, I can multiply top and bottom by two, that will give me two and I'll rewrite it a square root of X looks a little nicer square root of X plus X plus E. And there it is. That is the anti derivative of our given function. Okay, hopefully that helped to have a wonderful day.

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