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Evaluate the integral.

$ \displaystyle \int_0^\pi t \cos^2 t\ dt $

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Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 5

Strategy for Integration

Integration Techniques

Oregon State University

University of Michigan - Ann Arbor

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.

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.

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Evaluate the definite inte…

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Evaluate the integral.…

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Evaluate the indefinite in…

Let's evaluate this in a girl by using the integration by parts. Let's take unit B t Do you equals DT? Then we're left over with DV equals coastlines flair DT But here to find me we will have to do a little bit of work. You have to integrate co science where? Just use the half angle identity Here this is one plus coz I into t all divided by two Pull up the one half So here we have tea over to and then sign two tea over for if there's two teas bothering you, you could do a use up here and no need to add the constant in a veneration here because well, either it in at the very end. So now recall the formula integration by parts for a definite in a roll. So plugging in are you and B we have t square over too. If he signs of Tootie over four zero pie, that's two you times me and then minus in a girl syrup. I t over too Scientist t over for So let's just go ahead and evaluate this first expression over here by plugging in pie first and then zero So there you end up with Paice. Where Over too After simple Fine. And then over here for the Southern girl. So here that he So let's actually pull of the minus Then we'LL integrate So this will also change using the power rules he square over for And then this is we'll have co sign of two t over eight zero pie and then here this will be a minus So hear this minus turns this into a minus and the integral of minus sign is co sign positive Go sign. And that's exactly what we have here after this double minus. So go ahead and plug in pi and zero in for tea and we'Ll end up with Thais clear over four in this equals Pi ce were over for And that's our answer

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