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Problem 37 Hard Difficulty

Evaluate the integral.

$ \displaystyle \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \cot^5 \phi \csc^3 \phi d \phi $


$$\frac{22}{105} \sqrt{2}-\frac{8}{105}$$


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Video Transcript

this problem is from Chapter seven, Section two. Problem number 37 in the book Calculus Early Transcendental eighth Edition by James Stewart Here we have a definite integral from pi over Florida pi over two coz Hanjin to the fifth Power of feet times cozy Can Cube defeat. Since we have an odd power on the cozy can, let's go ahead. And Pelata Co Tanja and a cozy can't from the Instagram. So doing so we're left over with co tangent to the fourth, which we can we can write as co tangent Squared Square. So that's called Tangent to the Forest. Then we have cozy, can't squared and then we have our co tangent remaining co tangent Kosygin on the on the right side. Yeah. Next, we can go ahead and apply this path A grenade entity on the right to rewrite this co tangent squared and the parentheses circled in red now, so this becomes Kosik and squared minus one. And let's leave everything else as it is. So we still have cozy can square times co tangent times cozy can and the reason what we won't combine the cozy can square in the cozy can is because we'll eventually use a u substitution and this will be our d you. So now for the U substitution, we should take you to be cause I can't so that negative do you is co tangent times cozy again. T v. Also, because we have a definite integral, our limits of integration will change. So the new lower limit will become cozy can of power before push from the unit circle or using the definition of sign. You can evaluate this to be radical to, and the upper limit will be Kasey Kahne. Piper, too, which is one. So after this use substitution. We have negative integral Route 2 to 1 U squared minus one and the parentheses that's all being squared, and we're left over with U squared. And as we mentioned before, this is coming from our u sub. That's just R. D U. So let's go ahead and simplify this Inter grant as much as we can. Then we'll integrate. So here, if you don't like the minus sign, you could go ahead and using the properties of integration. You can push the negative sign. You can eliminate the negative sign as long as you switch the limits of integration. So I'll drop the minus. And we could put one here, two here. That's one way to proceed. We can go ahead and square this out each of the fourth minus two U squared plus one. All times you squared, and then we could go ahead and distribute this. You squared through the parentheses. So we have you to the six. To you to the fourth plus U squared. Do you? Yeah. Mm hmm. Let's go ahead and use the power rule three times here to evaluate these three other girls. So we're coming on over here. So we have you to the seventh power over seven minus two u to the fifth over five plus u cubed over three. And we have our own points. One route to right. So let's go ahead and plug these in. We plug in route to for you. Route two to the seventh. Two times. Group two to the fifth Power. So that's not a minus. Sign there needing me. Erase this. That's a radical two to the fifth. That's divided by five. Plus Route two to the third power over three. Okay, so that's when we plug in route to for you. Then we plug in one for you. We have 1/7, minus 2/5, plus 1/3. So you can go ahead and use your laws of exponents to evaluate. Simplify these radicals. And after doing so, you should get 22 radical to over one. Oh, five, 105 in the denominator. And for the second parenthesis that simplifies to 8/1. Oh, five. And here, if you'd like, you could combine that numerator. So here, let me take a step back. We have 22 route to minus eight, all divided by one oh five. And there's our final answer.