Problem 35 Hard Difficulty

Evaluate the integral.

$ \displaystyle \int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \cot^2 x dx $

Answer

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Video Transcript

this problem is from Chapter seven section to problem number thirty five in the book Calculus Early Transcendence, ALS eighth edition by James Store And here we have a definite integral from pi over six. The power to of co attention squared of X. So the first thing we can do is just use the definition of co attention to rewrite. It is cosign over sign. So have the integral from power sits the power to of coastline squared of X over science part of X And then we could apply a pathetic and identity and the numerator to rewrite co sign squared is one minus sign square So next begins lit up this fraction to write this is to separate in the rules. So for the first integral we have in a girl from however six the power too one over science clear And then we have minus the integral pi over six however two of just one because sine squared over Science square is just one. So let's make some remarks before we evaluate this first integral so we can rewrite this in a role by using the fact that well, first we can, right, This is equal to negative in a girl one over Science Square is just Kosi can square So we really haven't changed the integral Put a DX over here We haven't changed the integral because these negatives cancel out and won over Science squared is because he can't square. But by writing in this form, it might be easier to evaluate and sissy that this is negative coz the engine of X plus he because a derivative of CO attention is negative co sequence where seizing that fact we have negative attention, X And we're evaluating this at five or six in power too and that the integral of one, it's just next. And we're also evaluating that five or six and part of it too. So now we're readyto plug in the end points So negative Cho attentional power too minus negative co tension five or six. So this is the the first integral and then for the second animal we have a minus, however, to minus part over six. So for the first term, this negative co attention You could use the definition here too, right? This is negative co sign a pie over too over. Sign a pie over too and then we will plus co sign a pie over six over. Sign of five or six. Then we have a minus. So pi over two is three pi over six, minus pi number six. So if you want to five or six. So using the fax from the unit Circle, we know that cosign a pie over to a zero sign apartment. Who's won. So hear this. First home is a zero coastline of five or six using the unis circle again. Room three over, too and sign a power sixes, one house. And we could simplify this other fraction, too. Right? It is pi over three, and finally we could cancel the two's here in the first fraction, and we get well in three minus pi over three, and that's our answer.