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Use a graph of the integrand to guess the value of the integral. Then use the methods of this section to prove that your guess is correct

$ \displaystyle \int_0^{2\pi} \cos^3 x dx $

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$$=\int_{0}^{0} 1-u^{2} d u=0$$

Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 2

Trigonometric Integrals

Integration Techniques

Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

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.

06:22

Use a graph of the integra…

04:30

09:28

07:06

67-68 Use a graph of the i…

00:56

Evaluate the definite inte…

03:17

01:07

Compute the definite integ…

for this problem would like to use a graph of the Inter grand to guess the value than a girl. And then we'LL actually a computer and a girl to see if it matches our guests. So let's go to a graphing calculator and see what the graph of the thean so grand, which is Cho sang kyu defects, Looks like for X between zero and pi over, too, till here's a graph, of course, Thank you. And so the integral will be the area underneath the curve from zero to pie. So over here, from zero to about one point five we see or which, in other words, pie, we see that we have positive area. Then, from about one point five to four point five, we see a larger negative area, and then afterward we see more positive area. So my guess is that this that an area underneath the curve down here is twice the area of each of these positive sections. So my guess is that the area's cancel out and I will get zero. So let's evaluate this integral to see what we actually get so we can rewrite this as in a girl's era to Pie Co sign Squared temps go sign and then using a potato grin identity. We can rewrite coastlines. Where is one minus science? Where? So let's go ahead and do that. So have one minus sine squared times co sign and this looks like we should go ahead and do a U substitution. So let's go ahead and do that take you to be signed so that do you is co sign. Also, our limits of integration will change. So the lower limit will become sinem zero, which is still zero, and the upper limit will become sign of two pi, which is zero to a swell. So after you substitution, we have the integral zero zero one minus you squared to you. And since the lower and upper limits of the integral of the same, they're both zero here. That means that the answer will be zero, which is consistent with the guests from the graph. So we have zero, and that's your final answer

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