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By multiplying out the integrand, evaluate $\int\left(5 x^{3}+1\right)^{4} x^{4} d x.$

$$\frac{625}{17} x^{17}+\frac{250}{7} x^{14}+\frac{150}{11} x^{11}+\frac{5}{2} x^{8}+\frac{x^{3}}{5}+c$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 3

The Substitution Method

Integrals

Baylor University

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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01:24

in this problem, we want to find the integral of X. If it minus four x cubed plus one over X cubed minus four x dx. So in the numerator have 1/5 order volatile middle the denominator. We have a sword or on So before we can use partial fraction decomposition, we would want to use Long division or some other way to break this up. Let's go ahead and try. Long division still have x huge plus zero X squared minus four x divided into you, X If it close zero acts the fourth minus four x cubed about zero ax squared plus zero x plus one about the zeros, um, as placeholders to make it a little easier to organize. Okay, so first we wanna most bye bye expiry. So it'll be x to the fifth zero x to the fourth minus four x cubed. So from here, when we subtract that we get 000 plus one yet so we plus one over X cubed minus four acts. So with the remainder, we can perform partial fraction decomposition and then we'll have something we can probably integrate directly. So one over x cubed minus four X first. What we want to do is factor the denominator so we know what the factors are. So we factor in exile will get X times X squared and support and X My export minus four is the difference is Qari sweet and write that has X minus two times specs for us too. So now that we know what the factors are, we can set up our form. So have a over X plus B over X minus to plus C over exposed to you since all the factors or linear and on repeating, we can use the heavy side cover about the Sherman with undetermined coefficients are in order to do that. First we need to find the roots of those factors. So have X equals zero X equals two and X equals negative Seo. So then, if we're covering up X, we know that A is gonna be equal to one over negative two times two, which equal Teoh Negative one Force B is going to be equal to one so recovering of X minus two. So we have two times two plus two. So that's gonna be 1/80. And for see, that's gonna be. So we're gonna have one over negative two times native to minus two. That's gonna be equal. Teoh won over eight elderly founder factors. We can go ahead and rewrite the inner role. We have X squared minus 1/4 over acts plus 1/8 over X minus two plus 1/8. Overexposed Teoh DX. We can integrate all these directly, um, by definitions and rules. So have the power rule for the first term, So have X cubed over three minus 14 Helen Absolute value X plus 1/8 AL An absolute value of X minus two plus 1 ft l an absolute body of X plus two you plus a concept so we could combine these l in terms of we wanted Teoh. So let's go ahead and show how that works. So I excused over three. So, um, if I was to factor 1/8 from each of them this one, we get squared. As we have one case, Times two, we could pull that, too. In front will pull the two into the l N now would make that x squared. So that's gonna be plus 1/8 Ln absolute value of X minus two times exports that we could just rewrite that that was the difference of squares. Will have X squared minus four over X squared. Close the absolute value. We still have policy, right that finishes her problem.

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