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Evaluate the integral.

$ \displaystyle \int^{1}_{0} (x^{10} + 10^x)\,dx $

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00:47

Frank Lin

01:08

Amrita Bhasin

Calculus 1 / AB

Chapter 5

Integrals

Section 4

Indefinite Integrals and the Net Change Theorem

Integration

Missouri State University

Harvey Mudd College

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.

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we're doing the integral from 0 to 1 of X to the 10th. Power Plus 10 to the x power. Yes. Now you can google rules, you memorize rules, you can do a lot of things. But uh if you're familiar with the derivative of a to the X. Is equal to natural log of the base times eight of the X. Well the anti derivative or the integral is working backwards. So it makes sense if you have those rules already memorized divide as you go backwards. So as we're looking at this, we still have the power rule where we add one to the exponent and you multiply by the reciprocal of that. But this is exactly I'm talking about is a lot of students are used to going one way, we'll just go the opposite way. So divide by the natural log of 10 Times 10 to the X. And that's from 0 to 1. So now we're ready for this final answer because we can plug in our upper bound in for both of these excess and uh one to any power still once or do with that. Um And then 10 to the first power just be 10. Now you do need to subtract off plugging in zero and for both of these exits. Now this one is not a big deal because it is zero, But this one is a huge deal because 10 to the zero power is one Tend to the zero power is one. So as we're going to simplify this, We can combine the like terms that are dividing by natural log of 10. Um and it's 10 over natural log of 10 -1 over natural log of 10. So that would be nine over natural log of 10. And I believe this is your best answer. So we can circle it and move on.

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