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`int 10/((x - 1)(x^2 + 9)) dx` Evaluate the integral

Integrated `int10/[(x-1)(x^2+9)]dx`

Solve for the variables A, B, and, C using the method of partial fractions.

`10/[(x-1)(x^2+9)]=A/(x-1)+(Bx+C)/(x^2+9)`

`10=A(x^2+9)+(Bx+C)(x-1)`

`10=Ax^2+9A+Bx^2+Cx-Bx-C`

`10=(A+B)x^2+(C-B)x+(9A-C)`

Equate coefficients and solve for A, B, and C.

`0=A+B`

`A=-B`

`0=C-B`

`0=C+A`

`10=9A-C`

`0=A+C`

`10=10A`

`A=1`

`C=-1`

`B=-1`

`int10/[(x-1)(x^2+9)]dx=int[1/(x-1)+(-1x-1)/(x^2+9)]dx`

`=int1/(x-1)dx-intx/(x^2+9)dx-int1/(x^2+9)dx`

The first integral matches the form`int(du)/u=ln|u|+C`

`int1/(x-1)=ln|x-1|+C`

Integrate the second integral using u-substitution.

Let `u=x^2+9`

`(du)/dx=2x`

`dx=(du)/(2x)`

`-intx/(x^2+1)dx=-x/u*(du)/(2x)=-1/2ln|u|+C=-1/2ln|x^2+9|+C`

The third integral matches the form `int(dx)/(x^2+a^2)=1/atan^-1(x/a)+C`

`=-int1/(x^2+9)dx=-1/3tan^-1(x/3)+C`

`=int1/(x-1)dx-intx/(x^2+9)dx-int1/(x^2+9)dx `

`=ln|x-1|-1/2ln(x^2+9)-1/3tan^-1(x/3)+C`

...

Integrated `int10/[(x-1)(x^2+9)]dx`

Solve for the variables A, B, and, C using the method of partial fractions.

`10/[(x-1)(x^2+9)]=A/(x-1)+(Bx+C)/(x^2+9)`

`10=A(x^2+9)+(Bx+C)(x-1)`

`10=Ax^2+9A+Bx^2+Cx-Bx-C`

`10=(A+B)x^2+(C-B)x+(9A-C)`

Equate coefficients and solve for A, B, and C.

`0=A+B`

`A=-B`

`0=C-B`

`0=C+A`

`10=9A-C`

`0=A+C`

`10=10A`

`A=1`

`C=-1`

`B=-1`

`int10/[(x-1)(x^2+9)]dx=int[1/(x-1)+(-1x-1)/(x^2+9)]dx`

`=int1/(x-1)dx-intx/(x^2+9)dx-int1/(x^2+9)dx`

The first integral matches the form`int(du)/u=ln|u|+C`

`int1/(x-1)=ln|x-1|+C`

Integrate the second integral using u-substitution.

Let `u=x^2+9`

`(du)/dx=2x`

`dx=(du)/(2x)`

`-intx/(x^2+1)dx=-x/u*(du)/(2x)=-1/2ln|u|+C=-1/2ln|x^2+9|+C`

The third integral matches the form `int(dx)/(x^2+a^2)=1/atan^-1(x/a)+C`

`=-int1/(x^2+9)dx=-1/3tan^-1(x/3)+C`

`=int1/(x-1)dx-intx/(x^2+9)dx-int1/(x^2+9)dx `

`=ln|x-1|-1/2ln(x^2+9)-1/3tan^-1(x/3)+C`

The final answer is:

`=ln|x-1|-1/2ln(x^2+9)-1/3tan^-1(x/3)+C `

Are argon, oxygen, and water particles similar to neon particles? Why or why not?

First of all let's define the different kinds of chemical compounds we are talking about because that will define the kinds of particles we are talking about. Argon is an element on the periodic table. It is a member of the noble gasses, meaning that it is a monoatomic pure gas. In other words, argon gas particles are simply pure single atoms of argon. Oxygen is also an element on the periodic table but in nature, the simplest form of oxygen is oxygen gas, a diatomic molecule composed of O2 (two oxygen atoms bonded together). Since O2 is a diatomic molecule, its smallest particle size is a molecule. Finally, water is a polyatomic molecule H2O composed of two atoms of hydrogen and one atom of oxygen. Again, particles of water are the individual molecules.

Neon is also a noble gas just like argon. So neon particles are individual neon atoms. Neon particles are similar to argon particles but not similar to oxygen or water molecules.

What conflict do the bells and the wind present to the speaker in Stopping by Woods on a Snowy Evening?

When the narrator of "Stopping by Wood on a Snowy Evening" pauses on a dark, snowy night to watch the snow fall in the woods, his horse, accustomed to stopping in the town, is confused. "He gives his harness bells a shake/ To ask if there is some mistake." The horse is used to being driven from farmhouse to farmhouse, and he doesn't understand the reason the narrator stops in the dark woods to contemplate nature. The bells are the call back to the town and to human society. As the narrator pauses in the woods, "The only other sound’s the sweep/ Of easy wind and downy flake." In other words, all the narrator can hear is the wind in the woods. The wind is the call to stay in nature that is in opposition to the pull the narrator feels to go back to town.

After liquids, which expands most?

Matter expands in response to temperature due to an increase in kinetic energy—vibration—of molecules/ atoms. Gases expand the most, followed by liquids, then solids.

Expansion, more specifically thermal expansion, is the tendency of objects to expand—change in shape, area, and/ or volume, in response to a change in temperature.

An increase in temperature results in an increase in kinetic energy in a system, and this increase in kinetic energy of molecules translates to an observable...

Matter expands in response to temperature due to an increase in kinetic energy—vibration—of molecules/ atoms. Gases expand the most, followed by liquids, then solids.

Expansion, more specifically thermal expansion, is the tendency of objects to expand—change in shape, area, and/ or volume, in response to a change in temperature.

An increase in temperature results in an increase in kinetic energy in a system, and this increase in kinetic energy of molecules translates to an observable expansion of an object. Expansion can be seen as an increase in separation between atoms. This occurs because of an increase in vibration in them, caused by the increase in kinetic energy when the temperature was raised.

All matter, to some extent, undergoes thermal expansion—some more prominently than others. In general, the following gives the three phases of matter in order of increasing potential to expand:

solid < liquid < gas.

That is, gases expand more, followed by liquids, and then solids.

Gases have molecules that are already energetic. They move around and occupy every available space in a container/ vessel. Consider a balloon filled with helium. Increasing the temperature would cause the gas molecules to move faster, hitting the walls of the ballon harder, and ultimately expanding it. You can also see the reverse happening when you put a balloon into a freezer—you lower temperature, which lowers kinetic energy, and the volume decreases.

The same applies to both liquids and solids, although to a lesser extent. In both cases, atoms and molecules are closer to each other and have stronger interactions than in gases. The interaction is stronger in solids, making expansion less prominent. Expansion still happens, though. Train tracks have gaps in them (of a few centimeters) to allow for thermal expansion.

A line passes through the point `(10,-2)` and forms with the axes a triangle of area of 9 sq units. Find the equation of the line.

Hello!

Denote the slope of this line as `m.` The vertical line (which has an undefined slope) doesn't suit us, so we'll not miss a solution. Horizontal line with `m=0` doesn't suit also, so we can divide by `m.`

The equation of such a line is `y = m*(x-10) - 2.` The triangle formed with this line and the axes is a right one (because the axes are perpendicular to each other). So its area is `1/2 * |OX| * |OY|,` where `O` is the origin, `X` is the x-intercept of the line and `Y` is the y-intercept.

The y-intercept is `y(0) = -10m-2.` The x-intercept is the `x` for which `y= m*(x-10) - 2=0,` so it is `2/m+10.`

Thus our equation for m becomes

`1/2 |(-10m-2)*(2/m+10)| = 9.`

It is the same as `|(5m+1)*(1/m+5)| = 9/2,` or `|1/m (5m+1)^2| = 9/2.`

If we suppose `m` is positive, then it becomes

`(5m+1)^2 = 9/2 m,` or `25m^2+10m+1=9/2 m,` or `25m^2+11/2 m +1=0,` or `50m^2+11m+2=0.` This equation has no solutions.

Well, what about negative m's? The equation becomes `(5m+1)^2 = -9/2 m,` or `25m^2+10m+1=-9/2 m,` or `25m^2+29/2 m +1=0,` or `50m^2+29m+2=0.`

The discriminant is `D = 29^2-4*50*2 = 29^2 - 20^2 = 9*49,` so `sqrt(D)=3*7=21.` The solutions are `(-29+-21)/100,` `m_1 = -50/100=-1/2,` `m_2 = -8/100 = -2/25.` And both are negative as supposed.

Uff. There are two possible equations, `y=-1/2(x-10)-2=-1/2x+3` and `y=-2/25(x-10)-2=-2/25 x-6/5.`

What sort of evidence is there that the earth is or is not flat?

At this point, there is such overwhelming evidence that the earth is round and revolves around the sun that the only way in which it makes sense to discuss flat earth theories is in historical context.

Before the advent of modern technology and science, or understanding of gravity, many cultures considered the earth to be flat. Much of this was simply intuitive, due to the fact that it appears flat as we walk or ride horses or sail across short distances in sail boats. Although many Mesopotamian and other early cultures considered the world flat, as explorers traveled across larger distances and measured the positions of the sun from different latitudes, they began to realize that the earth was in fact spherical, but sufficiently large that its curvature was not readily observable by the naked eye. By 240 BC, the Greek scientist Eratosthenes had calculated the radius of the earth with a fair degree of accuracy, and increasingly accurate calculations of earth's radius and its distance from the sun became common in antiquity and have continued to be improved upon until the present.

The first circumnavigation of the globe was completed by Portuguese explorer Ferdinand Magellan in 1522 by ship. With modern technology, circumnavigation of the earth has become common, as have direct observations of a spherical earth from space.

Why does Bud always introduce himself as "Bud, not Buddy?"

Bud answers this question for readers in chapter five of Bud, Not Buddy.During this chapter, the narrative flashes back to a time when Bud's mother was still alive. Bud and his mother are having a conversation with each other, and she clearly explains to Bud why he is not "Buddy." She tells Bud that if she wanted her son called "Buddy," she would have named him exactly that. She then goes on to tell...

Bud answers this question for readers in chapter five of Bud, Not Buddy. During this chapter, the narrative flashes back to a time when Bud's mother was still alive. Bud and his mother are having a conversation with each other, and she clearly explains to Bud why he is not "Buddy." She tells Bud that if she wanted her son called "Buddy," she would have named him exactly that. She then goes on to tell Bud that "Buddy" is a name that people give their dogs. Her son will not be called a dog's name.

She'd tell me, "Especially don't you ever let anyone call you Buddy, I may have some problems but being stupid isn't one of them. I would've added that dy onto the end of your name if I intended for it to be there. I knew what I was doing, Buddy is a dog's name or a name that someone's going to use on you if they're being false-friendly. Your name is Bud, period."

Bud's mother gives a second reason for naming her son Bud. She tells Bud that a "bud" is a waiting flower. It is waiting to show the world its purpose and beauty. To Bud's mother, her son is like that flower bud. He's waiting to show the world how awesome he will be.

"A bud is a flower-to-be. A flower-in-waiting. Waiting for just the right warmth and care to open up. It's a little fist of love waiting to unfold and be seen by the world. And that's you."

Bud tells people that his name is "Bud, not Buddy" as a way to honor his mother's wishes. She gave him that name, and she was very clear that was what she wanted her son's name to be. Bud makes it equally clear to other people what his name is and what he wants to be called.