What drove Mark Twain to become a cub-pilot on a Mississippi riverboat?

In Chapter Four, Mark Twain tells us he always wanted to be a steamboat pilot; it was his childhood ambition for as long as he could remember.
He tells us that, as time progressed and all his boyhood friends grew up, many heeded the call of the river by working as ship engineers, mud-clerks (general workers who performed the dirtiest jobs on a steamboat), and pilots on the Mississippi River. Mark Twain asserts that a steamboat pilot's pay was in itself a great lure; every pilot was paid between $150 and $250 a month, a "princely salary" at the time. Additionally, the steamboat pilot never had to pay for board, and his salary easily eclipsed that of a preacher's. To become a steamboat pilot, one had to train as a cub-pilot. This is partly why Mark Twain was driven to become a cub-pilot; without the training, he would never have realized his ambition of becoming a steamboat pilot.
Mark Twain also liked the idea of traveling on the water; such an adventure exhilarated him and gave him a profound sense of well-being and vigor. On a ship bound for New Orleans, he wrote,

When we presently got under way and went poking down the broad Ohio, I became a new being, and the subject of my own admiration. I was a traveler! A word never had tasted so good in my mouth before. I had an exultant sense of being bound for mysterious lands and distant climes which I never have felt in so uplifting a degree since. I was in such a glorified condition that all ignoble feelings departed out of me, and I was able to look down and pity the untraveled with a compassion that had hardly a trace of contempt in it.

So, basically, Mark Twain was driven to become a cub-pilot because of a boyhood ambition to work as a steamboat pilot and a desire to reap the material benefits from such an occupation. Additionally, he also viewed a life on the water as one replete with adventure and excitement.

In "Three Men in a Boat" by Jerome K. Jerome, what are Montmorency's objections to the boat trip? Are they valid?

Montmorency is a fox terrier – the dog represented in the title, Three Men in a Boat (To Say Nothing of the Dog). By the end of Chapter I, the men have decided to take a trip along the River Thames. To be fair, the fourth member of the crew is granted a vote:

The only one who was not struck with the suggestion was Montmorency. He never did care for the river, did Montmorency.
“It’s all very well for you fellows,” he says; “you like it, but I don’t. There’s nothing for me to do. Scenery is not in my line, and I don’t smoke. If I see a rat, you won’t stop; and if I go to sleep, you get fooling about with the boat, and slop me overboard. If you ask me, I call the whole thing bally foolishness.”

So, the dog’s perceived objections are that (1) he would have nothing to do on the boat, (2) he would be bored by the passing scenery, (3) he was not one to sit back and smoke a pipe or cigar to wile away the time, (4) the boat would likely not stop when he would want to explore something, and (5) he could accidentally fall overboard while sleeping. A few of these concerns are well founded. Montmorency has nothing to do on the trip, and he may have been bored. He does not drop into the water while sleeping, however. He doesn’t anticipate any problems or interactions with the tea kettle, either. And when he does spy a water-rat, he is able to grab it and donate it to a project George undertakes in Chapter XIV. He does indeed participate fully in this journey, and he does stay with the three men for the course of their adventure. He may have even enjoyed the experience and may have forgotten his earlier protests, by the end.

How does the author portray the true destruction of war without a drop of blood in "Old Man at the Bridge"?

The story shows how war destroys and upends lives without having to shed blood. It opens with a picture of the dusty men, women, and children, along with their mule-drawn carts, displaced from their homes by the Spanish Civil war, crossing a bridge as refugees:

There was a pontoon bridge across the river and carts, trucks, and men, women and children were crossing it. The mule- drawn carts staggered up the steep bank from the bridge with soldiers helping push against the spokes of the wheels. The trucks ground up and away heading out of it all and the peasants plodded along in the ankle deep dust.

Being forced to flee from one's home, maybe never to return, or to return to ruins, is a way war can bring destruction into the lives of ordinary people.
The old man's worries about the fate of his vulnerable animals also shows the damage wrought by war. The man's cats may run away and his birds fly away from the cage he has left open, but to what fate? How will they survive? The old man must carry with him anxiety and pain about the well-being of the animals that he has been forced to leave behind in a dangerous war zone, saying:

A cat can look out for itself, but I cannot think what will become of the others.

The story shows how war reaches into and harms the lives of people and animals that are not a part of the fighting and have no idea what the carnage is about.

In Ernest Hemingway’s “The Old Man at the Bridge,” there is only a single moment of happiness, and even that is infused with violence and destruction. When the soldier asks the old man where he came from, the old man answers:

“From San Carlos,” he said, and smiled.
That was his native town and so it gave him pleasure to mention it and he smiled.

The irony, of course, is that this old man “with very dusty clothes” has been driven from San Carlos by the artillery. His concern is no longer for himself but instead for the collection of animals he was forced to leave behind during his evacuation. Although the old man is, by his own description, “without politics,” they are still upon him, and the violence they wreak is visible through both the language employed by the short story as well as the man's foreshadowed death.
“The Old Man at the Bridge” is an incredibly short story, and the dialogue that makes it up can be described as compressed. The bloodshed that exists just at the margins of the story has reduced the interaction between the old man and the soldier to something transactional. The two speak in the briefest of sentences, and it is rare for any spoken sentence to be more than six or seven words.
Ultimately, the soldier recognizes that the old man’s fate is more or less sealed. While bloodshed remains outside of the page itself, the final passage makes clear that the soldier believes that the old man’s death is impending:

There was nothing to do about him. It was Easter Sunday and the Fascists were advancing toward the Ebro. It was a gray overcast day with a low ceiling so their planes were not up. That and the fact that cats know how to look after themselves was all the good luck that old man would ever have.

Calculus: Early Transcendentals, Chapter 7, 7.4, Section 7.4, Problem 35

int (dx)/(x(x^2+4)^2)
= int 1/(x(x^2+4)^2)dx
To solve, apply partial fraction decomposition. So to express the integrand as sum of proper rational expressions, set-up the equation as follows:
1/(x(x^2+4)^2) = A/x + (Bx+C)/(x^2+4) + (Dx+E)/(x^2+4)^2
Multiply both sides by the LCD.
1=A(x^2+4)^2 + (Bx+C)(x)(x^2+4) + (Dx+E)(x)
1=Ax^4+8Ax^2+16A + Bx^4+Cx^3+4Bx^2+4Cx+Dx^2+Ex
1=(A+B)x^4 + Cx^3 + (8A+4B)x^2+(4C+E)x +16A
Express the left side as polynomial with degree 4.
0x^4 + 0x^3+0x^2+0x+1 =(A+B)x^4 + Cx^3 + (8A+4B+D)x^2+(4C+E)x +16A
For the two sides to be equal, the coefficients of the polynomial should be the same. So set the coefficients of the two polynomials equal to each other.
x^4:
0 = A + B (Let this be EQ1.)
x^3:
0=C (Let this be EQ2.)
x^2:
0=8A+4B+D (Let this be EQ3.)
x:
0=4C+E (Let this be EQ4.)
Constant:
1 = 16A (Let this be EQ5.)
Notice that the value of C is already known. So let's solve for the values of A, B, D and E.
Plug-in the value of C to EQ4.
0=4(0) + E
0=E
Isolate the A in EQ5.
1=16A
1/16=A
Then, plug-in the value of A to EQ1.
0=1/16+B
-1/16=B
And, plug-in the value of A and B to EQ3.
0=8(1/16)+4(-1/16)+D
0=8/16-4/16+D
0=4/16+D
0=1/4+D
-1/4=D
So the partial fraction decomposition of the integrand is:
1/(x(x^2+4)^2) = (1/16)/x + (-1/16x+0)/(x^2+4) + (-1/4x+0)/(x^2+4)^2 = 1/(16x) -x/(16(x^2+4))-x/(4(x^2+4)^2)
Then, proceed to take the integral of it.
int1/(x(x^2+4)^2) dx
= int (1/(16x)-x/(16(x^2+4))-x/(4(x^2+4)^2))dx
= int 1/(16x)dx - int x/(16(x^2+4)) dx - int x/(4(x^2+4)^2)dx
For the second and third integral, apply u-substitution method.
u=x^2+4
du=2x dx
(du)/2=xdx
= int 1/(16x)dx - int 1/(16u)*(du)/2 int (1/(4u^2) *(du)/2
= 1/16int 1/xdx - 1/32int 1/u*du -1/8int 1/u^2du
=1/16int 1/xdx - 1/32int 1/u*du -1/8int u^(-2)du
=1/6ln|x| -1/32 ln|u| +1/(8u)+C
And, substitute back u=x^2+4 .
=1/16ln|x|-1/32ln|x^2+4| +1/(8(x^2+4))+C

Therefore, int (dx)/(x(x^2+4)^2)=1/16ln|x|-1/32ln|x^2+4| +1/(8(x^2+4))+C .

The Things They Carried is a story about war. Do you think it is an anti-war story? Why or why not?

My answer to this question is yes. An author does not necessarily have to state explicitly that "war is wrong" or "war is stupid" in order to make an antiwar point. The reader can infer the writer's perspective on war from the incidents in a story and from the way the story is told, regardless of the lack of preaching or any specifically didactic approach to instruct the reader about the "message."
O'Brien's narrative through the interlocking stories of The Things they Carried is often understated and ironic. The lists of things of sheer weight, the physical implements the soldiers are carrying, seem endless. In the first story, itself titled "The Things They Carried," we are told that the men burn a village. It's stated matter-of-factly, as business as usual. Lt. Cross feels guilt when one of his men is killed, and dreams of his girl at home. But one senses that numbness has enveloped officer and soldier alike. The platoon smoke grass, as everyone in the war did, and move on. They discard items because they know a chopper will come along and easily re-supply them, because the U.S., waging this seemingly irrational war 10,000 miles away, is rich and bountiful.
In O'Brien's writings, as in most war literature from Crane to Remarque to Ron Kovic, the men in the field have no connection to the master plan behind the war or generally even to the reasons for the specific missions they are on. They go through the motions and do what they are told to do. It is not so much a conception of the "wrongness" of war as it is a theme of the impenetrable nature of war, of the motivations for it which the men in combat are charged with fulfilling. This is the essence of "antiwar" writing.

The Things They Carried by Tim O’Brien is an anti–Vietnam War story. The story details the mental and physical weight of the war. O’Brien never states outright that the story is anti-war, but he doesn’t need to do so. His descriptions help the reader understand the burden of being in a foreign land and fighting an unknown enemy. These men are literally and figuratively lost while fighting a war they don’t really understand.
The fighting seems secondary to the mental anguish the soldiers endure—most notably the fact that they could die at any moment. Simply put, the men have to fight if they want to survive. That being said, the men are not necessarily compelled to fight. O’Brien explains the soldiers’ greatest fear is the “fear of blushing. Men killed . . . because they were embarrassed not to. It was not courage, exactly; the object was not valor. Rather, they were too frightened to be cowards.” The story seems anti-war because we see troops grappling with the emotional baggage that comes along with going to Vietnam, instead of men fighting for glory or valor.

Before answering this question, it's important to understand that it's quite difficult to say anything definitive about Tim O'Brien's The Things They Carried. Indeed, the book's confusing structure defies efforts to define its meaning at every turn. That said, I think one could say that it is, in fact, an antiwar story, at least for the most part.
Consider, first of all, the book's chaotic structure. This structure has several effects. First, it mirrors the chaotic state of warfare in which the characters are embroiled. Jumping back and forth in time, revisiting certain events more than once, and generally lacking any kind of linear structure, the "plot" (if there is one) of The Things They Carried mirrors the confusing, chaotic, illogical nature of the Vietnam War. Moreover, it mirrors the psychological states of most of the work's characters. Faced with war and death, the soldiers in O'Brien's work experience the deterioration of their own mental landscapes. Norman Bowker, for instance, carries the thumb of a dead Viet Cong child and then, after returning from the war, commits suicide. Additionally, Rat Kiley, unable to deal with the death of one of the platoon's members, brutally tortures a water buffalo. Clearly, the characters in O'Brien's fiction are suffering and psychologically traumatized, and the fragmented, confusing nature of the author's prose reflects this notion.
With these points in mind, I think it's possible to say The Things They Carried is an antiwar story. By focusing on the chaos and the trauma, and by mirroring these realities in a fragmented, chaotic narrative, O'Brien skillfully illustrates the calamity of war and death.

Calculus of a Single Variable, Chapter 8, 8.6, Section 8.6, Problem 30

For the given integral problem: int sqrt(x)arctan(x^(3/2))dx , we can evaluate this applying indefinite integral formula: int f(x) dx = F(x) +C .
where:
f(x) as the integrand function
F(x) as the antiderivative of f(x)
C as the constant of integration.
From the basic indefinite integration table, the problem resembles one of the formula for integral of inverse trigonometric function:
int arctan(u) du = u * arctan(u)- ln(u^2+1)/2+C
For easier comparison, we may apply u-substitution by letting:
u =x^(3/2 )
To determine the derivative of u, we apply the Power rule for derivative:d/(dx) x^n = n*x^(n-1) dx.
du =d/(dx) x^(3/2)
= (3/2) *x^(3/2-1) * 1 dx
= 3/2x^(1/2) dx
=3/2sqrt(x) dx
Rearrange du =3/2sqrt(x) dx into (2du)/3 = sqrt(x) dx .
Plug-in the values u = x^3/2 and (2du)/3 = sqrt(x) dx , we get:
int sqrt(x)arctan(x^(3/2))dx =int arctan(x^(3/2))*sqrt(x)dx
= int arctan(u) *(2du)/3
Apply the basic integration property: int c*f(x) dx = c int f(x) dx .
int arctan(u) *(2du)/3 =2/3int arctan(u)du.
Applying the aforementioned formula for inverse trigonometric function, we get:
2/3int arctan(u)du=(2/3) *[u * arctan(u)- ln(u^2+1)/2]+C
=(2u * arctan(u))/3- (2ln(u^2+1))/6+C
=(2u * arctan(u))/3- ln(u^2+1)/3+C
Plug-in u =x^(3/2) on (2u * arctan(u))/3- ln(u^2+1)/3+C , we get the indefinite integral as:
int sqrt(x)arctan(x^(3/2))dx =(2x^(3/2) * arctan(x^(3/2)))/3- ln((x^(3/2))^2+1)/3+C
=(2x^(3/2) * arctan(x^(3/2)))/3- ln(x^3+1)/3+C
or (2xsqrt(x) arctan(xsqrt(x)))/3- ln(x^3+1)/3+C

Note: x^(3/2) = xsqrt(x)

Authors provided positive and/or negative critique of the readings.

A positive critique of the stories in Meeting at the Crossroads would be that the book takes an important issue and sheds light on it by tracking and examining how girls learn to transition into women and how it affects their ability to express themselves. Brown and Gilligan are able to drive the point home by interviewing the same girls year after year while being open and honest about their methods and limitations.
Meeting at the Crossroads by Lyn Mikel Brown and Carol Gilligan argues that women are forced to give up relationships with themselves, their bodies, and the people around them when they learn how to have romantic relationships. This leads to girls losing their voices. The girls tend to lose their individual freedom as they try to adapt to what society expects from women. 
One of the major strengths of Meeting at the Crossroads is that Brown and Gilligan talked to the same girls from before puberty until they were adults; they interviewed the girls from 6 to 18. This enabled them to see both how the girls changed and how the girls' relationship with the interviewers changed. As time went by, many of the subjects even came to question the motivations of the interviewers and what kinds of answers they actually wanted.
One of the positive things that Brown and Gilligan do in the book is point out their own limitations. They are aware that they might not be getting fully open answers and that they were potentially another cause of anxiety for the people they interviewed. The authors don't attempt to hide these issues. They're very open about them—which lends authenticity to the rest of their stories and findings.
Another interesting aspect of Brown and Gilligan's process for getting stories from the girls was that they changed the method they intended to use. At first, they intended to ask the girls the same questions at every interview. Instead, they ended up letting the girls lead the interviews so they could figure out what each girl valued. This is very different from traditional interview methods used by psychologists.
There are positive and negative stories in the text from each girl. One negative example is that of Victoria, who experiences abuse at home and tries to please the abuser. She fantasizes to the interviewers about having a healthy relationship and killing violent men. She then quickly retracts the statement, putting herself down. One positive example is Sonia, who blooms when she is able to be interviewed by someone of her own race. Brown and Gilligan see her experience as a sign that women may be able to help each other through the tough transition from child to adult.
By being open about the limitations of their research methods, the changes they made, and the methods they used, Brown and Gilligan are able to shine a positive light on the stories they collected from the girls—both from their words and their behavior or body language. Even the negative stories in the text contribute to the argument the authors are making, which makes their inclusion important.