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Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral. $ \displaystyle \int_0^{\pi} x^3 \sin x\ dx $

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$\pi^{3}-6 \pi$

Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 6

Integration Using Tables and Computer Algebra Systems

Integration Techniques

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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.

27:53

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

01:16

Use the Table of Integrals…

02:17

03:20

02:23

02:43

08:27

Use the Table of Integral…

03:14

03:28

Okay, This question asks us to compete the value of this integral. So one way you could do it is with a table, as they suggest. But since that involves a lot of Riker jh in, it might actually be easier to use integration by parts because we know how to do this. So we'll just use the d I method or the ladder method, whatever you want to say. So we have our d column here and there. I call him here. So our first term is X cubed, and then we just keep taking derivatives. Then for I column, we start with sine X and then we integrate negative coastline X negative sign X co sign X sine X and then we multiply the diagonals together, remembering that it goes plus minus plus minus. So we get negative. X Cube co sign X minus a minus. So plus three x squared Signed X Plus six ex co sign X minus six signings and we're evaluating this from zero to pie. So if we plug in pie, we get negative pi cubed times negative one plus zero plus six pie times negative one plus zero minus. So this is our upper limit and then our lower limit is well X cube zero x squared to zero access zero and sign X zero. So zero. So our final answer turns out to be just negative. Pi cubed times in negative one plus six pie times negative one, which is hi cubed minus six pie And that's our final answer. And again, I just used the D ay method because, quite frankly, I think it's much faster than plugging into the table over and over again. But you're more than welcome to do that if the D ay method does not appeal to you.

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