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Evaluate the given integral.$$\int x^{5}\left(3 x^{6}-2\right)^{9} d x$$

$$\frac{\left(3 x^{6}-2\right)^{10}}{180}+c$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 3

The Substitution Method

Integrals

University of Michigan - Ann Arbor

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.

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the derivative of the inside function, which is three times X to the sixth minus two is contained in this integral um um which is X to the fifth. And all we have to do is take you to be the inside function. That way we can get rid of X to the fifth d X. So if we take you to be three x to the six minus two, then we get Do you is equal to 18 times X to the fifth D X, and 18 times next to the fifth. That's not quite X to the fifth, but we can just divide through by 18 to get d you over 18 is equal to X to the fifth d X. Now we can remove X to the fifth DX, replacing it by d'you and remove three Xa six minus two. Replacing that by you rewriting this integral as 1/18 times the integral of you to the ninth. And this is equal to 1/18 times you to the 10th over 10 plus C, which is equal to one over 180 times three x to the six minus two raised to the 10th Power plus C, and that completes the problem

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