By dividing the range into 4 equal parts . the tabular forn for sin* dx is given below0.707Find the missing values (green colour) in the table3I "=0.041 at * =...

Establish the indicated value of the given definite integral. $$ \int_{0}^{1} \frac{\sin x}{x} d x=1-\frac{1}{3 \cdot 3 !}+\frac{1}{5 \cdot 5 !}-\frac{1}{7 \cdot 7 !}+\cdots $$

For this problem we are asked to establish that the integral from 0 to 1 of sine of X over x dx equals one minus 1/3 times three factorial plus 1/5 times five factorial minus 1/7 times seven factorial. So the first thing that we can do is note that we can expand out sine of X in its MacLaurin series as X minus X cubed over six Plus X to the power of five over. um Actually I'll be careful about how I say this. Actually we can write that as X cubed over three factorial plus extra power of 5/5 factorial minus X to the power of 7/7 factorial. Having that if we then divide through by X we just change that actual one and reduced down the exponents of each of these. So if we're trying to integrate from 0 to 1 sine of X over X D X. And that will be the integral from 0 to 1 of each one of those terms. Plus some leftovers. So one goes to X and then minus X squared over three factorial. We'll go to minus X cubed over three times three factorial. I'm going to pause because of ambulance. I don't think I've gone a single day of recording without being interrupted by an ambulance. All right, so we have plus X para 4/4 times four factorial minus X. The power of 7/7 times seven factorial evaluated from 0 to 1. So when we do the evaluation we are left with one minus 1/3 times three factorial plus 1/4 times four factorial -1/7 times seven factorial.

Arrested in degrees zero to pi of sine three x co sign for X dx. And I'm going to use the formula forgiveness for this, which tells me that the integral of this is co sign of and minus end. So three minus four X divided by few times and minus at which is three miles, four a minus the co sign of M plus ends of three but for divided by two times 34 and then we're going from zero pie. Okay, so that gives me no sign of mine. Divided by three minus four is minus one times to his mind it and then my it's go time seven divided by 14 I'm plugging in zero and high, So I do that many a co sign of minus I divided my mind to mind of seven point before I mean And then Meyers the co sign Rome. You my divided by four. Okay, so what is the co sign of minus pi minus one over minus two minus the coast side of seven. Pie is also my one or working. And then it's minus 01 minus one and then mine. It's among excuse me a plus the hotel as you grow it again once 1/4. Okay, so what does that mean? Well, let's your over signs. We're gonna get 1/2 plus 1 14 uh, plus 1/2 plus 1 40 So what does that give me? Well, I get I put them all over 14. I get 1/2 is 7/14 plus one plus seven plus one. So that is the poodle of well, over 14 and 12 or 14/7. My six cents. I just need to correct something, though, because I am looking at this one more time, and that should have been negative. So that should be a negative, which means that this should be negative. And this should be negative. Which means that when I put in this, it's actually positive. And this is going to be positive, which means I'm going to have, Let's see, minus one, have ah, minus 1/2 which is minus seven and minus seven, which is minus 12 which is minus 6/7. That's my fault. And so near Rio. I just miss read the formula from the textbook, so it should be a mine is in front of that 1st 1 That's my fault. And I'm sorry with that. Although I'm recognizing now that I added 7711 and got 12. Which is disappointing, actually. But when you have minus seven minus seven, you had one. And when you get minus 12 so it all works out In the end, the answer is, um there you go. Two wrongs make a right.

So not to integrate Sanders to four X -3. We know we cannot integrate function of function integration directly. So we need to convert this into a single function. So here we have signed these two folds we can write this a science square X -3 Whole Square reason for doing this. Since we know formula for science care which can what it into a without square time. So what is that? We have 1- course to later is Science square to Science Square theater. So if we apply this here, sorry We'll have 1 -4 two times x -3 square. Now again we have this is linear but we have square terms we have to apply against square formula. So u minus Behold square Is a Square -2. A Just two X -6. Be it is one plus b square. Sorry this was being this is a B squares which is caused square of two x -6. Now we are done with this one. We can solve this one but again here we have course and square. So again we'll apply half angle formula for that. So this is one minus. Or let me integrate separate the integration the integration of one day x minus two times integration of course two x -60 x Plus here. Now this course one plus cost to theater is to cause theater to cause square theater. So here since we have to change this course Square trita in terms of this so we need to divide by two years, we'll have to buy this and similarly years. So I mean To divide here too and on squaring will get four so that can be taken out as one x 4. Okay so let's put it here one plus course two theaters. So we have two x minus 6 to 9. This will become four x -12 x two Dicks. And here we have one rifle So now we can solve our integration. So integration of one is X -2 times integration of course is signed two x minus six. Now here X is multiplied by two. So we need to divide by two Plus. Here one x 2 will come out Integration of one is X plus integration. Of course signed four X -2. Well here X and multiple waterfalls. We need to divide by four and this is our solution. If you take one x 4 inside we'll have X x four. This get cancer -1 Way for sign two X -6. No here this is one x 4 times half. That becomes one x 8 x And this becomes one way 8432. Sign for x minus two. It and finally we need to add see So this is our solution but sometimes can be added here. So expire four plus one by Or x by eight is 3 X by eight. Then we have -1 by food Plus one x 32. Bless you. So there's a final answer. Thank you.

It's another one of those problems where you have two different trig functions and two different angles. So you need to know one of those product formulas which I don't know. So I have to generate them every time. I do know the sign of A plus B. And I know the sign of A minus B. And I need to use those because they have signing co sign in each term. And then when you add them together you get to sign a sign be so divide by two. There's the formula that I'm going to use an A. Is three X. And B. S. Four X. So this turns into zero to pi Sign of Add the two Angles 7 X. Plus. Um Subtract the angles sign minus X. All over to D. X. Okay now the sine of X looks like this. Okay so notice I drew the part in the first quadrant and then I flipped it upside down and drew it in the second quadrant. That means that the sign is odd. Okay so the sign of minus X is the opposite of the sine of X. Sign of negative X. Is the opposite of the sine of X. All right so that the opposite of the sine of X. So that means I can get rid of this minus sign here. So I'm gonna put the one half here is zero to pi signed seven X -1. DX. Okay because they are opposites now I can integrate in a grill of the sign of seven X. I need a seven and a 17 So 17 minus 1/7 coastline seven X minus minus cosine X. From 0 to Pi so one half -1 7th. Cosign seven Pie Plus coastline. seven pie -17. Cosine of zero Plus coastline of zero. Okay seven pie. So here's 1234 56 seven. So 7 Pi is the same as pie. Excuse Me -10 is the name of this point. And remember the X. Value is the coastline. So we got one half -17 times minus one. Oh Yeah this should not be seven pi this should just be pie right here because I put it here that's where the this one comes from and then I put it here which is where this one comes from But it is the same. So we get a -1 there minus -1 7th and mhm. Um the coastline of zero is 1 so that we get all that one half 1/7 minus one minus minus 1/7 plus one one half of six minus 6/7 minus six sevens. Yeah so that's minus 12 7th Divide by two So -6 7th.