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Evaluate the integral.

$ \displaystyle \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \csc^3 x dx $

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Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 2

Trigonometric Integrals

Integration Techniques

Missouri State University

Campbell University

Oregon 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.

03:06

Evaluate the integral.…

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Evaluate the definite inte…

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Evaluate the given definit…

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07:15

this problem is from chapter seven, Section two Problem forty of the book Calculus Early. Transcendental. Lt's a tradition by James Door and we have a definite integral of Kosi can. Cute. So let's start off by doing integration. My parts Let's take you too, because he can So that do you is negative. Cosi again. Time to pay attention, D X and buy a choice if you were left over with TV, which has to be cause he can't you, Ursus is because you can swear, Deeks. So that be is negative attention of books. Okay, so we comply. Are under grace. My parents Were you tansy. So negative. Cozy again. Times call attention here and points are five or six three minus the integral of video notice here were negative on the and also on to you. Those will cancel out. So we just have co attention. Times co tension times co seeking so cozy engine square because he can. So if you think so, observe here. First we see in this integral we have co tension square so we can apply our protagonist identity over here and read on the right to rewrite this as cause he can't squared minus one. So let's go ahead and replace CO. Attention squared with co sequence four minus one. Let's also go to the side and simplify this cause he can cho tension and evaluated at these end points. So let's call this jay So we have J equals negative cause he can't power. Three. Attention Preparatory minus Cho Seon over six. Cho Attention five or six and going in and using our knowledge from the unit circle, we can evaluate each of these to over three one of our route three minus two times three, which we could simplify as two three minus two over three. So that's a J. So that's over here. Circle and green. So picking up where we left off, we have James, which is now two three, minus two or three minus. And now for our new into rule, we have Kosi can square times. Costigan is cozy, can cute, which you would have noticed that's original problem, and then Kosi can X times minus one is negative. Kosi can IX. So since we see the original integral popping up again, let's go ahead and to know this original problem, it's integral by eye so that we see that here. Once again, we have eye again. So's Gordon. Move this eye to the other side. But let's be careful here because we also have a negative Cosi kit and the integral. So we have two I two three minus two over three. And then we have a minus minus. So plus integral Piper six number three Kosi convicts D X. So for the next there, we need to discuss this integral Cosi convicts. So first, let's just fight the previous equation that we had. We have two I two three minus till Rhodri. What now this in a girls in your table's in the section so we can go out and use that fact to evaluate this inaugural. So we have two or three minus two of the three. Then we have minus natural log Absolute value Cho tension of X plus Cassie Innovex evaluated at the end points however six and territory. All right, so this is still too I So then now we're ready tto softer isolates divide both sides by two. So we have We're three minus one over three and then we have minus one half and now we plug in pi over three and five or six. And Tex So plugging in power three first. And then we subtract when we plug in power for six. So here we can go ahead and evaluate. And we have natural log of excuse me, one over. Radical tree, plus two of a radical theory. And again here, um, pushing the minus sign through the apprentices. So that's a plus natural log. And then I have a one happens. Well, radical three plus two. And excuse me. There was, ah matter, mistake here. That should have been a rule three. And we could go on and clean this up a little bit. So here we have Ellen of three over room three bushes, just room three and again using some properties of the logs. You could even simplify this latest expression if you wanted to, but otherwise, this is our final answer.

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