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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 \sin 2 \pi x \cos 5 \pi x dx $

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Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 2

Trigonometric Integrals

Integration Techniques

Campbell University

Oregon State University

Baylor 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.

09:28

Use a graph of the integra…

03:09

06:22

05:57

Evaluate the integral.

03:17

Evaluate the definite inte…

02:55

08:29

05:19

Evaluate the indefinite in…

04:19

Evaluate the integrals.

07:38

Evaluate the given definit…

01:18

Evaluate the integrals by …

here we'LL use the graph of the immigrants, I guess the value of the integral. Then we use the methods of the section toe. Check her answer. So here our Into grand is signed to PI acc's Times Coast on five packs. So let's go to the side and bring this up on the graphing calculator. So here's the graph of the Inter grand from zero to, so it's a bit tedious looking, but we observe that we have one portion of the graft between zero and one. And then if you flip this upside down room, so reflect this about the X axis and then we should get this the part portion of the graft between one and two. So it looks like you just obtained the second part of the grab I reflecting the first part about the X axis. So because of that, my guess is that the area will be zero. So let's see if that's actually the case. So because our Interbrand is of the form, sign a co sign, be weaken, use the formula and this in the textbook in this section that lets us rewrite us. So we have sign of a minus B plus sign of a plus B, all divided by two. And our problem is to pi x Bia's five Pious IX. So it's quite and use this formula. We have one half was pulled that fraction zero to then we have a sign and then a minus. B is negative three packs and then a plus B seven packs. Now we can evaluate each of these inner worlds using a use of if you like here for the first one, you could take you to be negative three packs and here you could take you two be seven pious IX. So after integrating, we have one half and a girl of sign is negative co sign. So we have a negative co sign. Negative three packs. Overnegative three pie minus co signed seven Pi X over seven Pie and ER, and points are from zero to notice. Here we can cancel off these minus signs, so it's ignore those now we're ready to plug in the end points, so it's a plug into. First, we have co sign of negative six by over three pie minus coastline of fourteen pie over seven pie. So this is from plugging into now we'LL go ahead and plug in zero Cosa no. Zero was one. So we have won over three pie minus one over seven pie, So evaluate. We know. Co sign of minus six pie and co sign a fourteen pyre. Both won. So we have one half one over three pie minus one over seven pie. Then after we distribute apprentices minus one over three pie plus one over seven pie and we see here, we could do some cancellation. So we have one half time zero, and we get zero, and that's your final answer.

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