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Evaluate the given integral.$$\int\left(3 x^{5}+1\right)^{8} 15 x^{4} d x$$

$$\frac{(2 x+1)^{10}}{10}+c$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 3

The Substitution Method

Integrals

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

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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Evaluate the integral.…

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09:32

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Evaluate the given integra…

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to solve this integral, we can use the use substitution because the derivative of this inside function three acts to the fifth plus one is equal to 15 X to the fourth, which is right here. So if we use a use substitution, we can bring out the 15 X to the fourth. And we can replace three X to the fifth plus one by you and solve this integral, um, using power rule. And that will be much easier than multiplying all these terms out. So let's go ahead and take our use substitution by allowing you to be three X to the fifth plus one. That way, do you is equal to 15 x to the fourth DX, and now we can rewrite this integral. So this is the integral of you to the eighth Power times. Do you? Right, Because do you replaces 15 x to the fourth d x. So now when we integrate this, we get that this is you to the ninth over nine plus c. And now we can replace you with three X to the fifth plus one and that completes the problem

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