5

Evaluate the definite integral:3 10x+ 9 dx3 + 9dx1Ox(Type an exact answer; using radicals as needed:)...

Question

Evaluate the definite integral:3 10x+ 9 dx3 + 9dx1Ox(Type an exact answer; using radicals as needed:)

Evaluate the definite integral: 3 10x + 9 dx 3 + 9dx 1Ox (Type an exact answer; using radicals as needed:)



Answers

Evaluate the integrals.
$$\int \frac{3 d x}{\sqrt{1+9 x^{2}}}$$

We have the integral from 3 to 5 of x times square root X squared minus nine DX. Now, notice that the derivative of X squared minus nine is two X. So effectively, all we need to do is undo the power rule on the square root X squared minus nine. So we could do X squared minus nine, raised to the three halves. And then we're gonna multiply it by these things up front to make it work. Okay, so now the coefficients simplify to one third. Okay? Okay. And we get 25 minus nine. Race to the three halves and zero. So this is one third times 16 3 halfs, which is 64 3rd.

Let's have value the integral of the square root of X squared minus nine over Execute. This is a perfect candidate for a truth sub. So looking at the numerator inside the radical, we see that we have an expression of the form X squared minus a square for history. And when you have ah expression of this form, you could try the truth Substitution X equals a c can't Data and Sensei's three. Our criticism will be X equals three seeking Then we can differentiate. So take the differential on the side d X equals three. See Canberra time stamp data debater. So before we go ahead and rewrite the integral in terms of they don't let's just go to the side and deal with this numerator So let's go ahead and simplify this first. So plugging in r x, we have nineths, he can't square is X squared minus nine So you can factor out tonight in here and take the nine two squared and nine outside. That becomes the three sequence four minus one and we know that this is tension squared on the inside. So this just becomes three. Ten data so we can rewrite are in a roll as the integral of three Tan theta and D X. We found that over here that's three times C can't time, Stan so separate this from our earlier work. So that's our numerator. Our denominator is X cubed. So let's twenty seven seeking you. So let's go ahead and cancel out as much as we can. We have a nine and the numerator so we could cancel let off with a nine down here and well looked over with a three. So the plot of wonder from the integral we have tangents where up top and we could cross off this. He can't with one of those in the bottom, and we have two left over in the denominator. So now let's go ahead and use the definition of tangent. That's science flared over co sign squared, and then we know that C Can is one over coastline so we can write sequence where, as one over sea cans flared as co sign squared, then we could cancel off the coastlines, and we have one third integral of science square. And since I'm running out of room, let's go on to the next page. So let me rewrite. Let's pick up where we left off. We had one third integral of science square, So now we have a triggered a metric in a rule. This is something that we've seen in seven point two, and since there's no co signs president, it's sine squared here. It will help to use the identity, that science, where is one minus coastline to data all over. Then we could go out in your memory. So first, maybe let's clean this up a little bit. Let's pull up the the two from the denominator. And now we can integrate this. We have won over six. So one day becomes data. And in a girl of coastline to data scientist, they'd over, too. Plus he and here it will help Tio when we because we're going to want to get everything back in terms of X, it might be difficult to find sign of tooth data in the triangle. So before we draw the triangle, let's use the double angle formula to rewrite this. We have won over six to six out their fate on minus to side data co signed data. That's a double angle formula for scientific data, and we still have this two on the bottom. Those will cancel. So is it. Right now we have won over six fate of minus signed data cosign data. Plus, he let me rewrite the previous expression. Now, this is where we can go to our original trick substitution. We're ready to get everything back in terms of X. So first we take our trick sub. We can rewrite this as seek and data equals X over three. This tells us how to draw the triangle, so C can of data is high pound news over the adjacent. So we have X over here. It's three down there. The missing side was calling H and then let's use put that reindeer on to find h h flared plus three squared is X square. So this means H is the square root of X squared minus nine. Now we have all three sides of the triangle and we could find scientist and co signed data. You have won over six Seita. We don't really need the triangle to find data. You could actually just take your shrinks substitution and soft later over here. So here we could take in verse. Ekin on both sides and we're left with data equals he can't inverse of eggs over to me, so that because that's our data value, so seek an inverse Exhibit three. Now for sign we have our triangle with no sign is opposite over hypotenuse so each other ex. So we have X square minus nine over Lex and then for co signed. That's a Jason over the hype on you. So three over x for co sign. So three over X was for co sign and scientist turned into this green expression. So the last thing to do is to just multiply out the one over six and to simplify. So we have seek an inverse of X over three, all divided by six minus. Now, this three over sixes, one half. So we have a two under denominator. We also have expired. And then we have this world on the top, and then we have to add our constant again. So this is our final answer

To find the anti derivative of this integral. The first thing we have to do yes to rewrite this into The integral of three over nine times x squared over nine Plus one. And then DX, which is the same as The integral of one over three times the square of x over three. And then plus one. And then DX. Next we will do substitution. You want to let U equal to X-plus three and you will be one third dx. So then this will give us the integral of D. U over U squared plus one. That's equal to tangent inverse of you. And then plus C. And since you is X over three, this is equal to Tangent inverse of X over three and then plus C.

So we have this here. Okay, We're gonna try integrate this one without trying to use any of the techniques of integration. How do you do that? The first thing here to notice is that this one is like a semicircle. Okay, so if you have those, uh, thing right here, this is like a semi circle. Okay? And that is that is the behavior of this function right here. Okay. Uh, my semicircle is not drawn to scale. Okay. And then this is the interval. Negative three and positive three. Okay, so it stretches negative three here and positive ve. Okay. So you can see here. Negative to your negative one k. Your positive to positive one your negative for, you know, negative. Positive. For So this is roughly how the function is looking like Kate and integral between negative three and positive three. It's just the area under the curve. That is a CNN definition of integration. Okay, so, uh, with that, what is that? The area under the curve. This is just the area of a semicircle. Okay. What is the area of a circle? Is pi r squared? So the area off a semicircle is just pi r squared over two because it's in my circle. It's just 1/2 off a circle. Right? So, pie. This year, what is what is it? Raises. The radius is just three. Okay, Radius is any life from the center of the circle to the circumference? Okay. And that is three surpise times. Three squared over two. Okay, so this is just nine by over two way Just violated this one without using any techniques.


Similar Solved Questions

5 answers
(6) (3, -2n/3)21(r, 8)) ( > 0)(r, 0) =(r < 0)
(6) (3, -2n/3) 21 (r, 8) ) ( > 0) (r, 0) = (r < 0)...
5 answers
Your answer is incorrect,point) Water is leaking " out of an inverted conical tank at rate of 14700.0 cm% Imin at the same time that water is being pumped into the tank at a constant rate_ The tank has height 8.0 m and the the diameter at the top is 5.0 m. Ifthe water level is rising at rate of 19.0 cm/min when the height of the water is 3.0 m, find the rate at which water is being pumped into the tank in cubic centimeters per minute.Answer: 0.0017cm" min
Your answer is incorrect, point) Water is leaking " out of an inverted conical tank at rate of 14700.0 cm% Imin at the same time that water is being pumped into the tank at a constant rate_ The tank has height 8.0 m and the the diameter at the top is 5.0 m. Ifthe water level is rising at rate o...
5 answers
Point) Solve the systemT1 +82 ~313 381 +4xz -2835~681T2+813
point) Solve the system T1 +82 ~313 381 +4xz -283 5 ~6 81 T2 +8 13...
5 answers
Law and determine the indicated final condition Identify the appropriate named ga5 101 K to201 K, determine the fnal volume: A 1.00 balloon Is healed (romcooled from 1041 {Cto 41 !C, delerina Ine Iinal pressun qayala pressure 15 mm HglIb gample pressuro o/ 00 alm Is comprossod to 1,0 L, dotermine (he final pressure A 12.0 L lube ol qas &t
law and determine the indicated final condition Identify the appropriate named ga5 101 K to201 K, determine the fnal volume: A 1.00 balloon Is healed (rom cooled from 1041 {Cto 41 !C, delerina Ine Iinal pressun qayala pressure 15 mm HglIb gample pressuro o/ 00 alm Is comprossod to 1,0 L, dotermine...
5 answers
For the following exercises, indicate whether each of the following statements is true or false. If the statement is false, provide an examole in which it is falseIf $b_{n} geq 0$ and $lim _{n ightarrow infty} b_{n}=0$ then $sum_{n=1}^{infty}left(frac{1}{2}left(b_{3 n-2}+b_{3 n-1}ight)-b_{3 n}ight)$ converges.
For the following exercises, indicate whether each of the following statements is true or false. If the statement is false, provide an examole in which it is falseIf $b_{n} geq 0$ and $lim _{n ightarrow infty} b_{n}=0$ then $sum_{n=1}^{infty}left(frac{1}{2}left(b_{3 n-2}+b_{3 n-1} ight)-b_{3 n} igh...
5 answers
Tnlo Quonton:8 0f 8 (0 coiplele) "Determnine whether the functiontan(ry) if (x,y) + (0,0) f(x,y) = Ty- if (x,y) = (0,0)is continuous at the origin
Tnlo Quonton: 8 0f 8 (0 coiplele) " Determnine whether the function tan(ry) if (x,y) + (0,0) f(x,y) = Ty- if (x,y) = (0,0) is continuous at the origin...
5 answers
Draw the Lewis structure for HCN. Indicate the hybrid orbitals, and draw a picture showing all the bonds between the atoms, labeling each bond as $sigma$ or $pi$.
Draw the Lewis structure for HCN. Indicate the hybrid orbitals, and draw a picture showing all the bonds between the atoms, labeling each bond as $sigma$ or $pi$....
5 answers
04: If each one of the ropes will break when it is subjected t0 tensile force of 450 determine the maximum uplift force F the balloon can have before one of the ropes breaks Q5: The thin ring can be adjusted vertically between three equally long ables from which the 100-kg chandelier is suspended. If the ring remains in the orizontal plane and the tension in each cable is not allowed t0 exceed determine the smallest allowable distance required for equilibrium.
04: If each one of the ropes will break when it is subjected t0 tensile force of 450 determine the maximum uplift force F the balloon can have before one of the ropes breaks Q5: The thin ring can be adjusted vertically between three equally long ables from which the 100-kg chandelier is suspended. ...
5 answers
Paul and Mary toss a fair coin in turn until one of them wins the game by getting the first "head." Calculate for each the probability that he or she wins the game.
Paul and Mary toss a fair coin in turn until one of them wins the game by getting the first "head." Calculate for each the probability that he or she wins the game....
5 answers
Is a European down-and-out option on an asset worth the same as a European down-and-out option on the asset's futures price for a futures contract maturing at the same time as the option?
Is a European down-and-out option on an asset worth the same as a European down-and-out option on the asset's futures price for a futures contract maturing at the same time as the option?...
3 answers
3 A lift can carry a maximum of 650 kg: Suppose that the weight of a person is normally distributed with expectation 75 kg and standard deviation 10 kg: Let Zn be the total weight of n randomly selected persons_Determine the probability that Zs 650. (b) Determine n such that P(Zn > 650) 0.01 .
3 A lift can carry a maximum of 650 kg: Suppose that the weight of a person is normally distributed with expectation 75 kg and standard deviation 10 kg: Let Zn be the total weight of n randomly selected persons_ Determine the probability that Zs 650. (b) Determine n such that P(Zn > 650) 0.01 ....
5 answers
4+5x Evaluate the integral: 50 I+r dx
4+5x Evaluate the integral: 50 I+r dx...
5 answers
Snlve the equation (x in Tdi4s and 0 in degrees) (0r Jll exact solulions whcre Ppropriate; Round Pproximate AnGTERE rdizns Io [Our decimal plates and approximate answers degtees Lle nEurellenin c852 x -Lanz
Snlve the equation (x in Tdi4s and 0 in degrees) (0r Jll exact solulions whcre Ppropriate; Round Pproximate AnGTERE rdizns Io [Our decimal plates and approximate answers degtees Lle nEurellenin c852 x - Lanz...
5 answers
A psychologist conducts a 2 x 3 x 2 ANOVA. How many main effectsare possible? How many interactions are possible?
A psychologist conducts a 2 x 3 x 2 ANOVA. How many main effects are possible? How many interactions are possible?...
5 answers
Sercmhee Queston sin? Vi Submit Queston the Tind 21 Submt Queston 8 te 20 Submi Queston 8 [egion sds indefinite area Jo enclosed tber integral 'U0iza} 4 and522 Decide whether to 1 Go1 with pt 02 respect to 0 Details3 pt 8 0 Details
Sercmhee Queston sin? Vi Submit Queston the Tind 21 Submt Queston 8 te 20 Submi Queston 8 [egion sds indefinite area Jo enclosed tber integral 'U0iza} 4 and 522 Decide whether to 1 Go1 with pt 02 respect to 0 Details 3 pt 8 0 Details...
5 answers
Ehat 1 is occupicd by 0.227 mol of COz at 288.3 K 1 895 mmHg?
ehat 1 is occupicd by 0.227 mol of COz at 288.3 K 1 895 mmHg?...

-- 0.019584--