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Evaluate the integral.

$ \displaystyle \int t \sin t \cos t\ dy $

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Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 5

Strategy for Integration

Integration Techniques

Oregon State University

Idaho State University

Boston College

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.

02:41

Evaluate the integral.…

03:04

00:24

Evaluate the indefinite in…

01:23

Evaluate the given indefin…

04:13

01:06

Let's use integration. My parts for this one. Let's take you to BT So that do you equals DT and then we're left over with Devi equals sign times co sign DT then the You could find this by just integrating. And this in a girl you've seen earlier in the chapter, you could do a use up here or let's use different letter w equal society. And here we'LL get science Where Over too And as usual, when using integration my parts Let's not add the constant C right now. Well, add that in the very end. So here, using the integration by parts you have beauty minus integral video Long has put Are you Indian? So this is our you and Harvey and then minus and a girl. And now we have V again and then do you? So we pull out that to outside the integral in half and then we just have signed square. Now for this you'LL use the half ingle formula If the right is one minus co sign to tea all over too. So let's go ahead and write that. And so combining those twos we get one over four in general one minus cose. Ianto T. Now there's two here. If that's bothering you, you could do another use up here. Let's, um, he's a different different letter here instead of you. Since it's already being used here, you could do another u sub and we should get T Thanks. Word over to still minus. And now we have one over four with the minus, and then that integral of one becomes a T. So that's minus t over. For here. We have a double minus that becomes a plus, and then we have one over four signed t over two plus e. Finally, we got our constant of integration and the last step here isjust combined just a multiply those denominators out. So now we have plus sign to t over eight plus e, and that's our answer.

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