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Evaluate the integral.

$ \displaystyle \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sin \theta \cot \theta}{\sec \theta}\ d \theta $

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Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 5

Strategy for Integration

Integration Techniques

Missouri State University

Baylor University

University of Michigan - Ann Arbor

Idaho State University

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.

02:43

Evaluate the definite inte…

02:35

Evaluate the integral.…

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09:50

Let's just start off here by rewriting this integral. So pining for six pi over three, and then we have sign and let me go headed instead of writing one over C camp. Just replace that with co sign and then apply the definition for coach engine that's co sign over sign and then deviate a still. Now let's go ahead and cancel those signs. We have co sign squared and then apply the half angle identity to rewrite Close Science cleared this up here. That's our co sign squared. And now we're using the Half ing Identity, which is replacing co sign squared with one plus co sign to Dana and then all divided by two. And we pulled out the one half of me. So go ahead and integrate this state over to Sines, who data over two times two Plug in those end points tops Piper three. So plug in pi over three first and then minus. And now plug in pi over six. And then here. These values will cancel out there the same, but we're surprising. And then you have pie over six minus pyrrhus. Well, fascist Positive pi over twelve, and that's your final answer

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