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Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral. $ \displaystyle \int e^t \sin (\alpha t - 3)\ dt $

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Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 6

Integration Using Tables and Computer Algebra Systems

Integration Techniques

Missouri State University

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

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.

01:29

Use the Table of Integrals…

09:15

Use the Table of Integral…

03:33

02:04

01:28

02:30

03:56

$5-32$ Use the Table of In…

00:52

Use the table of integrals…

02:59

05:30

Okay, so this question wants us to evaluate this integral. So to do that, we want to put it in the form of one of our integral Sze in the table. So we would like to make this sign of Alfa Times something so well, it's factor out an Alfa from our sign. And now let's call that Are you? So you is equal to T minus three over Alfa. So that means that d U is equal to D T and T is equal to you plus three over Alfa. So now we can do some back substituting and get the integral of e to the T, which is you. Plus three over Alfa Times. Sign of Alfa Times, you do you. And then as a last simplification, we can pull this e to the three Alfa outside of are integral sign due to exponentially ation loss. And now we're left with this, which is exactly one of our forms in our integral table. So if we go over there, we find this formula and in this case, are a is just equal the Alfa and we have ah e to the three over Alfa outside. Still Okay, so now we gotta back substitute times. Well, you he to the u So eat of the U is T minus river Alfa over one plus Alfa squared. And that is all times Sign of our original argument minus Alfa Co sign of our original argument, plus C. And now we can see one final simplification. So this e to the three over Alfa cancels with this E to the minus through Alfa. So then we get our final answer off E to the T over one plus Alfa squared times Sign of Alfa T minus three minus alfa Co sign of Alfa T minus three plus C And this is our final form of our answer.

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