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JH
Numerade Educator

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Problem 17 Easy Difficulty

Evaluate the integral.

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

Answer

$$\frac{1}{4} \pi^{2}$$

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Video Transcript

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