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Evaluate the given integral and check your answer.$$\int\left(4 x^{3}-9 e^{x}+\frac{8}{x}-5\right) d x$$

$$2 e^{t}-\frac{3 t^{5}}{5}+7 \ln |t|-12 t+c$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 1

Anti differentiation - Integration

Integrals

Campbell University

Baylor University

University of Nottingham

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…

02:44

Evaluate the integral.…

00:56

evaluate the following int…

01:50

03:47

01:06

Evaluate the indefinite in…

01:47

Find the indefinite integr…

02:06

Evaluate the definite inte…

03:56

Evaluate integral

eso were asked to find the anti derivative the integral of four Execute minus nine e to the X plus eight over X uh, minus five city DX eso What I would examine is the fact that the derivative of I'll say some constant times natural law Globex, we learned earlier is equal to see over X looked very similar to this thing as we work backwards where the council would be eight. The other thing that I would look at is that the derivative of some constant times e to the X just a different constant is equal to K E to the X, Um and that's gonna help us identify this piece where the constant would be negative. Nine. In that case, eso just a few more things were already ready for the anti director of you. Add one to your exponents and you divide by your new experimental four divided by forward canceled s so there's no need to write a one there. I guess if you wanted to, you could this an iterative would still be nine e to the X based off my work here. Um, now the one thing I do need to mention is that when you do, the derivative of natural gets implied that excess positive. But we don't know what this excess positive. So we put in an absolute value, so make it eight natural log of absolute value of X and then the anti derivative of five is five X. I'm just thinking about how the derivative of this gives me negative. That's how I think of it. I know other people work it out slightly different, and then don't forget about a plus C because you need a constant in here. We just don't know what that constant would be. And you check your work by taking the derivative of this and you'll see you get the same answer is what's up here. So we did this correctly.

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