Problem 60 Hard Difficulty

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 $

Answer

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Video Transcript

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.