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2x2 dx 2 Vx 251516. Find the average value of the function f(x)-2sin 2(x)cos' 2(x) on the interval [0,n/2]. Do the integration symbolically, then check with yo...

Question

2x2 dx 2 Vx 251516. Find the average value of the function f(x)-2sin 2(x)cos' 2(x) on the interval [0,n/2]. Do the integration symbolically, then check with your calculator.

2x2 dx 2 Vx 25 15 16. Find the average value of the function f(x)-2sin 2(x)cos' 2(x) on the interval [0,n/2]. Do the integration symbolically, then check with your calculator.



Answers

Find the average value of the function on the interval, using antiderivatives to compute the integral. $y=3 x^{2}+2 x, \quad[-1,2]$

So objective here is to find the average value of this function on the specific interval here. And we're going to be using the girls to do that here. So We have from 74 We'll do one of our P -A, which is 4- cereal here. And so then we have to put in the function Air which is C. Can't squared X T X. Okay, so in order to find the integrator of this, we're basically saying who's derivative gives us C can't squared X. The only answer to that is another trick function Just actually 10 FX. So that means that we have tan of X From 0 to 4 And then for 1/4 minutes, one, which is 1/4. And then we just plug in from there. So we have 1/4 tangent for this 1/4 attention of zero. Nice thing about this is that tension of 00. So it's just zero times 1/4. Just zero. It's her answer is 1 4th attention for

Using the average value formula for anti grills here and so we're going to be doing that on this interval from here to be here. Israel P is performed search function here, Serena do it from 0 to Pi before man. The average value would be one over B -A. Which is power for zero. So that's C can't square deck sticks. Okay, so now to find the anti derivative of this, we just have to ask ourselves who's ready to gives us. He can't squared of X. So the answer to that is tanja rex. So then with that in mind you have the integral of sequence critics, detention X. Taking that for zero to power for and subtracting area of one of her power for here. Yeah, Times two result that So now tangent of pie before That's equivalent to one. It's gonna be minus attention to syria, Attention to zero is single 20. So then power for once are in serious. One of the pie before is equal to four pi over pipe. That's her answer there.

Find the average value of the function y equals three X square plus two x on the interval. Negative 1 to 2. So here's the formula for the average value. Why with a bar over it? So why Bar is equal to one over B minus a, which is the length of the interval times the integral A to B. Why D x. Okay, so this a to B Y d X, that's the area under the curve y on. We're dividing it by the length. So if you have area divided by length what, you're going to get us height, which will be what you're looking for. Average value here. So we have boy prime equals one over tu minus negative one integral negative 1 to 2 parentheses. E three X square plus two x the X. All right, Don't forget to put this DX on here because it tells you that what you have here is an area. This is the high, and this is the width. So you have the area of one little, um, interval sub interval. And then this says, find all of those guys and I had them all up. Okay, so this DX is doing something, so don't leave it off. So you got one third. Okay? Now we're gonna do the anti derivatives here. So we have three times the anti derivative of X Square, which is executed over three plus two times the anti derivative of X, which is X squared over two from negative wanted to. So notice those threes cancel. And those twos canceled. Okay, so now we have one third times x to the third plus x squared, the minus 1 to 2. So one third time's two Q plus two squared, minus negative. One parentheses, cubed plus negative. One squared. So one third, eight plus four minus negative. One plus one negative. One plus one. That zero. So we get one third times 12, which is equal to four. Shit.

In discussion we are required to find the value of integration. Dx upon 225 16 after Squire with the help of table of integral. So let's see how to solve this question We know that 225 is the square of 15 and 16 is the Squire of four. Therefore the given expression can ultimately it is integration dx upon 15 square minus four X. To the power to no consider um vehicles to four X. Therefore by the differentiation we can ride Do you is close to four D. X. Hands do yes will be close to Do you upon four now substitute you on the place of four X. And do you upon four on the place of dx So we get integration. Do you accept on 225 miners 16 Access query the calls to integration burn upon 15 square minus the U. S. Choir into. Do you buy 4? Hence This will be called to one upon four integration do you upon 15 square minus you Squire. And then all from the table of integral. Apply the formula integration. Do you accept pawn a square minus access square? Is the calls to one upon two way log a plus X divided by a minus X plus constant of integration. So when we compare the left hand side of this formula with the obtained expression so we get a recalls to 15 and act physicals to you substitute all the values in the formula so we get integration Do you x upon 225 miners 16 x squared equals two. One upon four. Multiplied by one upon to it is 15. Yeah log 15 plus. You divided by 15 minus you plus constant of integration hands this will be calls to one upon 120 log 15 plus. You divided by 15 minus you plus constant of integration. Now substitute four x on the place of you. So we get integration. Dx upon 225 -16 X. Esquire it recalls to Went up on 120 log Yeah 15 plus four X upon 15 minus four X plus constant of integration. And this can also be written us integration. Do you accept on 225 minus 16 X squared equals to one up 120 log four x plus 15 divided by four x minus 15 plus constant of integration. So this is definitely answer for this problem. I hope you understood the solution. Thank you.


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