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Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral. $ \displaystyle \int_0^1 x^4 e^{-x}\ dx $

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$24-\frac{65}{e}$

Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 6

Integration Using Tables and Computer Algebra Systems

Integration Techniques

Missouri State University

Baylor University

University of Nottingham

Lectures

01:53

In mathematics, integration is one of the two main operations in calculus, with its inverse, differentiation, being the other. Given a function of a real variable, an antiderivative, integral, or integrand is the function's derivative, with respect to the variable of interest. The integrals of a function are the components of its antiderivative. The definite integral of a function from a to b is the area of the region in the xy-plane that lies between the graph of the function and the x-axis, above the x-axis, or below the x-axis. The indefinite integral of a function is an antiderivative of the function, and can be used to find the original function when given the derivative. The definite integral of a function is a single-valued function on a given interval. It can be computed by evaluating the definite integral of a function at every x in the domain of the function, then adding the results together.

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In mathematics, a technique is a method or formula for solving a problem. Techniques are often used in mathematics, physics, economics, and computer science.

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Use the Table of Integrals…

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Use the Table of Integral…

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Okay, This question wants us to evaluate this integral. So what we're gonna d'oh is these are integral table to solve this, but that formula could be kind of complicated. So we're just going to use the d ay method using integration by parts. So, in the d column, we just write our polynomial and keep taking derivatives and then all the way down to our final answer before a zero. And then for the eye terms, we just keep integrating those. And now we just puller ladder and what we need another one on this side. So we have this one's plus this one's minus this one's process ones minus this one's plus, and the final one is minus. So that means our anti derivative is next to the fourth e to the minus X. But we have a negative one here, minus for ex cute eat of the minus X minus 12 X squared Eat of the minus X, and you see what's happening here. Every term is a minus, and we're going from 0 to 1 so we can fact you're out our e to the minus X and get a simpler looking anti derivative still going from 0 to 1 and this gets us e to the minus one times negative one minus four minus 12 minus 24 minus 24 plus zero. Sorry plus minus eat of the zero times zero plus zero plus zero plus zero minus 24. So then all we have to do is simplify these because eating zeros one. So we'll just get eat of the negative one times this sum. So 24 minus 24 negative, 48 minus another 12 is negative, 60 minus another five is negative. 65 minus zero is one times negative. 24. So we get our final answer of 24 minus 65 divided by e.

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