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7 4 16 On 3Techniques 4 jo i x)'dx Value T)x f Reccnt Iatcgration Find the @Verage...

Question

7 4 16 On 3Techniques 4 jo i x)'dx Value T)x f Reccnt Iatcgration Find the @Verage

7 4 16 On 3 Techniques 4 jo i x)'dx Value T)x f Reccnt Iatcgration Find the @Verage



Answers

In Exercises $7-16,$ find formulas for the functions represented by the integrals.
$$
\int_{4}^{x} e^{3 u} d u
$$

In discussion. We need to evaluate the given definite integral which is four Rds over scheduled for minus S scared for the limit 02 I want so we can write this integral as I it calls to four times Integral from the limit 0- one. And uh run over scare route 14 billion is too scared minus as he scared dot Diaz Now we will use uh the formula for integration as integration of integration of run over Scaramucci is scare minus S. S care or sorry excess care dot dx is it calls to signing worse X over A plus C. So here we will use this integral and we will get the integral I. E. Calls to four times the integral of this will be sign in verse as over two. And for the limits 02. What now by executing the limits we will get this integral. I calls to four times signing verse one minus sine inverse. So you won by two Here as as we will substitute as as one And sign in verse zero and we know the sign universe one x 2 will be by by six and signing a zero will be zero. So this integral I will be called to four times Sign investment but it will be by by six And by solving this we will get this integral is To buy over three. So integral. I will be To buy over three. So we got the integral we need to find in discussion. I hope all of you got discussion. Thank you

Okay. We're looking to find a formula for this interval from two decks of unit before do you? In other words, we just take the integral plug in the bounds and simplify. So when is the anti derivative? Every year to the four. Well, I know it would have to be you to the five over five. Since when you take a derivative of that, the five comes down, Castor's with the five, and you get you to the four. So I know that's the anti derivatives, and now we have to plug me in X and two. Okay, Well, if we plug in X, that's simply extra five over five. Then we subtract what we get in the plug into, which is just true to the five to run it by five. And I know that two of the five is 32 divided by five, and that is it.

Now we're ready to do double integration, so we're going to integrate the same function first, with respect to y and then with respect to X and these problems, you can see that it's important that you write down D y and D X at the end or dx dy y because you got to know which variable is the variable for that piece. Okay, so first winners integrating with respect to why so that means six X squared is a constant integral of why? Why squared over two minus two X and then a y and thats 0 to 2. Okay, so you should go ahead and plug these in before you integrate anymore. So you get two squared for 2 12 x squared, minus for X, and then if you plug in zero, you get zero for those pieces. Now integrate with respect to X. So you get 12 x cubed over three minus four X squared over two from 1 to 4. So this is four times for to the third minus two times four squared 4 16 64 to 56 minus 32 222

Before we find the definite integral that question asks for, we need to find this indefinite integral sweets. You have just removed the balance. And ah, since it's a rational function here, we're gonna try the partial fractions. But that's over first just going to look at the function itself. So I'm gonna remove the integral sign and, um uh, now we do. We check that it's an improper fraction, which it is, and, um, the denominator is already fully factored so we can move on to the decomposition, um, in the denominator. And we just have three linear factors. So we're just gonna be putting, um ah, different constant over each of those factors. So a over why waas be over, Why was too plus c over Y minus three. Now, where we need to find a D and C are by something in different values of why, um, but first, it'll be awful. Teoh, get rid of the denominators by multiplying by this part on both sides. So we get, we get him. So you see it multiplying by white times. Wife was two times why my three on both sides and when you cancel out and everything we four y squared minus seven Y minus 12 equals a uh Why Plus two. That's why they're, um times. Ah, why my story must be Why? Why? Minus three plus c? Why? Why? Plus two. And now we will some values in for why first to find a We're gonna sub in Weigel zero so that this turn gets canceled out and this gym has canceled out. As you will see, I feel a if we let y equals zero, then, um then we'll see what we end up with for this equation. First, we have four times zero squared when seven times zero minus 12 equals a times zero plus Teoh times. Ah, zero minus three. And then, as I said, since why is equal to zero these But what if these trains get cancelled out? This is all we have. And of course, these first two parts are zero and zero. Then this is just two times negative three. So we have that we two times a year three is negative. Six. So the solution to this equation is a equals to there's the first constant. Now I was gonna shrink this so that we have more space. Now we're going to find the value, be by letting ah, like, well, negative too. So that this turned gets canceled in this turn gets canceled. And when we do that, I'm gonna line Teoh separate the word. So we're gonna let y equal needed to. And, ah, when we do that, we have four times Negative. Two squared, minus seven times negative too. Minus 12 equals. Um, so the aides that the A term and the A C turn canceled out. So we have b times negative too. Times a negative. Two minus three. Simplify this four times and I give to square to 16. Now you have seven times negative news 14 minus 12. This is equal to 10 B. The left side would be equal to 18 and therefore be equals 18/10 or nine over five. And, um lastly, we're gonna find Ah, see? So we're gonna let why equal three set both Eastern's. He canceled out. And when we do that, we have throwing away. Why equal three and then we get four times three squared minus 73 minus 12 equals. And then, um, the A and B terms get cancelled out so yet we're left with a C. See time's wife times why I'm opposed to or C times three times three plus two is five. So there, now we simplify. Four times three squared is four times nine, which is 36 36 minus 21 minus 12. People's 15 c Um, so three equals 15 C c equals point over five. Um, now we have our three constants and we can move forward from here. So now what we know is on the that indefinite integral that we were looking for, which I'm going Teoh, I'm gonna move that function from the top to the bottom so we can work with it. So, um, we found that a is equal to two. B is equal to, um 9/5. Si is equal to, um 1/5. And, um, that means the that indefinite integral we're looking for This is an immediate equal to be into indefinite integral of this. So for that indefinite girl, we just need to find this integral and ah, we can do that easily, knowing that that general integral of ah one over X is line absolute value X, and that would mean? So we take the integral of two over Why? Which is to find absolute value. Why, then we have into growth nine or five over. Why was he so 9/5? Bon absolute value. Not white was too. And then absolutely sorry. Integral of 1/5. Over. Why? Ministry so 1/5 1 absolute value. Why my history Notice? We don't need an arbitrary constants, since we'll be finding a definite integral anyway. And, um, that definite integral we were looking for. It's from 1 to 2 of that original function. Now, we just found that the indefinite integral as this integral without the bounce, is equal to this. So we need to do is to find this is sub are upper bound into this expression, and then I will lower bound into this expression. So, um, first we have we're gonna sub to in, so we have to line absolute value. Eso in place of why will put to you plus 9/5 blonde absolute value. Uh, too close to you. Plus 1/5 mon absolute value to my three. Then, um we will subtract. And, um, I'm gonna move on to the next alliances were in outer space. And then we said, the lower bounce too long an absolute value. Uh, one US 9/5. Long and absolute value. Uh, one plus two plus went over five. Lawn, absolute value. Um, one minus three. Now, you just need to simplify this, uh, looks this number. So I left 212 Absolutely. Twos. Two plus 9/5. Um, absolute value. Say line of absolutely to Pristina just for, um, plus. Ah, well, one of, uh, absolute value to my history is lawn of one, which is zero. So we don't need to put that, Burt, This is just zero. And then, um, same applies to this part here. 110 So, uh, what's next is we can expand the negative to the brackets. So we have, uh, So you have minus nine over. Five one. Absolutely. One steel is three. And then, um, plus, plus lawn everyone, we're expanding. The natives minus minus 1/5. Absolute value. Sorry, Lawn. Absolute value on my street or 12 And now, um, we can simplify this further. Um, now, from here, we want to see, so this is technically equal to the integral. However, we want to simplify it even further. Notice that in this line we have four and four is the power of to which is in this line and this lawn. So we're gonna rewrite this. Ah, as a lot of to eso we can possibly combine it with the other wants. So we have two on two plus 9/5, and then four is to two out of two. So we have 9/5 times two or 18 over five. So 18/5 onto that's what 9/5 or four would be equal to. Then we're gonna move one minus 1/5. I want you over here next to it so we can combine the like germs and finally minus 9/5 13 And, um, now we can just, uh we have three terms that have long to we can add their coefficients to plus 18/5 minus 1/5 would've would add up to when you have the fractions. B 27/5 going to, um, and 27/5. That would be the same as nine times three. So it took 27 will be nine times three and we have minus 9/5 on three and by lawn rules or log rules. 9 9/5 times, three long to is the same as 9/5 types, lawn of two to power free or Lawn eight. And then we have minus 9/5 on three. So notice now we have a single efficient, too, so we can use log rules to combine these, and we get our final insert that the integral is equal to 9/5 times line of using log rules. 8/3. Ah, that's one final answer. However, we could also input in the calculator and get ah, an approximate answer of 1.765 therefore, are finally answer is, um, 9/5 Lawn 8/3, which is approximately equal to 1.765


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