Single Variable Calculus, Chapter 5, 5.5, Section 5.5, Problem 4

Find the integral $\displaystyle \int \frac{dt}{(1 - 6t)^4}$, by making $u = 1 - 6t$

If $u = 1 - 6t$, then $du = -6dt$, so $\displaystyle dt = \frac{-1}{6} du$. And


$
\begin{equation}
\begin{aligned}

\int \frac{1}{(1 - 6t)^4} dt =& \int \frac{1}{u^4} \cdot \frac{-1}{6} du
\\
\\
\int \frac{1}{(1 - 6t)^4} dt =& \frac{-1}{6} \int ^{-4} du
\\
\\
\int \frac{1}{(1 - 6t)^4} dt =& \frac{-1}{6} \cdot \frac{u^{-4 + 1}}{-4 + 1} + C
\\
\\
\int \frac{1}{(1 - 6t)^4} dt =& \frac{-1}{6} \cdot \frac{u^{-3}}{-3} + C
\\
\\
\int \frac{1}{(1 - 6t)^4} dt =& \frac{u^{-3}}{18} + C
\\
\\
\int \frac{1}{(1 - 6t)^4} dt =& \frac{(1 - 6t)^{-3}}{18} + C
\\
\\
\int \frac{1}{(1 - 6t)^4} dt =& \frac{1}{18 (1 - 6t)^3} + C


\end{aligned}
\end{equation}
$

When was the Civil War?

The American Civil War occurred in the mid-19th century.  It was a war fought between the northern Union states and the southern Confederate states.  The war was fought over a period of four years.
The Civil War began on April 12th, 1861 with the Battle of Fort Sumter.  Leading up to this battle, many southern states had succeeded from the Union.  Others followed after the battle and the start of the war.  Many people on both the Union and Confederate sides thought that the war would be a short one.  The Union had a stronger and more established army and more resources, but the Confederates fought with determination.
The Civil War raged on for years.  On April 9th, 1865, General Robert E. Lee found his army surrounded by Union troops.  He was forced to surrender at Appomattox Court House in Virginia.  News of General Lee's surrender took weeks and even months to reach other Confederate military leaders.  This was due to fragmented networks of communication.  President Johnson declared the official end of the war on May 9th, 1865.  The last Confederate general to surrender did so on June 23rd of that same year.

What might have happened in England and her North American colonies had the Glorious Revolution not taken place?

The Glorious Revolution of 1688 took place when the British Parliament invited the Protestant William of Orange and his wife, Mary, to take over Britain and depose the Catholic king, James II. In return, William and Mary agreed to give up some of the king's prerogatives or rights in favor of more power for the Parliament. This ended the period of instability in the monarchy and resulted in increased freedom for the American colonies. While the British monarchs strengthened their control over Ireland, they basically allowed the American colonies a greater degree of freedom that has been termed "salutary neglect." As a result, the colonies were accustomed to this freedom and reacted negatively when increased debt after the French and Indian War of 1754-1763 caused the British king, George III, to try to tighten his grip on the colonies and control their trade.
If the Glorious Revolution had not occurred, England might have continued to wrestle with religious strife related to the monarchy (as the Glorious Revolution resulted in a decree that the English monarch could not be Catholic). The Parliament might not have become as powerful as it did in Britain. In addition, if the Glorious Revolution had not occurred, the colonies may not have started the American Revolution, as they would not have been used to such a great degree of political and economic freedom. 

Are all three types of irony used in "By the Waters of Babylon"? Please give examples if they are.

Dramatic irony is when the audience knows something that characters in the story do not. We as readers know that John, the narrator, is describing New York City, and we know that in his visions he sees subways, airplanes, and nuclear war. John, coming from a far less technologically advanced culture, cannot understand as we do what he is seeing, and must describe it terms of his own context.
Situational irony arises when a situation is the opposite of what it seems. John and his people think gods built the big cities that are now called Dead Places, but we, as part of the civilization he is describing, know they were built by anything but gods. John will come to understand at the end that men built the cities, but he will not understand the full irony of the fact that the so-called "gods" destroyed themselves with nuclear weapons. Rather than be worshipped, our people should serve as a warning.
There is not much verbal irony in the story as John takes everything he sees with great seriousness and reverence, but there is some irony in George Washington's name being reduced to the letters ASHING, when his levelheadedness might have prevented a nuclear war and the ashes it produced.

There is definitely dramatic irony in "By the Waters of Babylon." Dramatic irony is a device that authors use to create situations where audiences know more about a situation than the characters do. 
In "By the Waters of Babylon," this kind of irony begins appearing once John arrives in the Place of the Gods.  We read about "god-roads," "SUBTREAS," and "ASHING." Those details allow readers to begin realizing that John is wandering around a fallen, post-apocalyptic city. Details continue to emerge about cold and hot water piping and the Grand Central Terminal. John is incredibly naive about what he is seeing, but readers eventually realize John is in is New York City. We are able to deduce that some kind of nuclear holocaust destroyed the human population, but John is completely unaware of all of this until the very end of the story. After he has his vision, then John realizes the gods were really regular people.   

Who would the antagonist be in "A Piece of Steak"?

At first glance, it may seem as though Sandel, the fighter who opposes Tom King, is the antagonist of the story. Roughly two-thirds into the story, it even states 

The third round began as usual, with Sandel doing all the leading, and delivering all the punishment. 

However, even though Tom King does not fight back, it turns out Sandel is not the true antagonist of "A Piece of Steak," as evidenced by this quote a little bit further on:

His gloves had already been removed, and Sandel, bending over him, was shaking his hand. He bore no ill-will toward the man who had put him out and he returned the grip with a heartiness that made his battered knuckles protest.

"A Piece of Steak" is atypical of many stories because it does not have a person who is the antagonist; instead, the antagonist is simply old age. In the beginning of the story, Tom King sees Stowsher Bill crying in the dressing room and cannot fathom why such a prized fighter would be bent over in tears. 
As the story continues, however, it becomes evident that Tom King is no longer in his prime. Though he was once the best fighter in the world, age and time have caught up to him. Toward the end of the story, it states:

King did not attempt to free himself. He had shot his bolt. He was gone. And Youth had been served. Even in the clinch he could feel Sandel growing stronger against him. When the referee thrust them apart, there, before his eyes, he saw Youth recuperate.

This shows that it doesn't really matter who the opponent is—youth will always prevail. The fact that the word "Youth" is capitalized throughout the story shows that it holds a place of reverence, and Tom King realizes that he has lost his somewhere along the way. It is his turn to become the once-prized fighter, bent over in the dressing room in tears. 

In Oedipus Rex, which concept—fate, karma, or luck—was used in the play? Explain how in detail.

Oedipus's path through life is controlled by fate. His queen, Jocasta (who is really his mother), heard from her now-deceased husband, Laius, that their child would grow up to slay his father and marry his mother. To prevent this fate, Jocasta and Laius abandoned their child when he was young. However, fate is decreed by the gods and cannot be averted or avoided though free will or any action taken by humans.
Oedipus, who was raised by another couple, also hears of his fate (that he will kill his father and marry his mother) from the oracle of Apollo. Not believing the oracle, he murders a man on the way to Thebes at the crossing of three roads. The man he murders turns out to be his father, Laius, though Oedipus does not know this. Oedipus goes on to marry Jocasta, also not knowing that she is his mother. When a servant arrives at the court to tell Oedipus and Jocasta that Oedipus was an orphan who was brought to his adoptive parents from the court of Laius, Oedipus and Jocasta realize that Oedipus's fate has come true was inevitable. Jocasta kills herself in response, and Oedipus blinds himself. Luck has nothing to do with Oedipus's outcome, as his entire destiny has been ordained. There is no element of chance. In addition, karma, or earning one's fate, is not relevant. Instead, Oedipus's fate was decreed at birth, and it is not a result of his actions. He has not earned his outcome; instead, it is his fate.

Yann Martel says in his introductory note, "This book was born as I was hungry." What sort of emotional nourishment might Life of Pi have given to its author?

The opening line of Martel's introduction says, "This book was born as I was hungry," but the next line says, "Let me explain" (vii). Martel then describes how he faced failure as a writer in the 1990s. This failure left him hungry for success for a very long time. As with any dream that seems to fail, a person will either give up or fight harder to achieve it. Failure also breeds discouragement, which Martel describes as follows:

". . . the whisper that has been pestering you all along from the back of your mind . . . speaking the flat, awful truth: it won't work" (viii).

Martel further explains that he felt as though an element was missing from his work. He compares this missing element of his story to a missing ingredient that makes a particular recipe work just right. Because of this missing ingredient, the author says that his story felt "emotionally dead," and it left him "with an aching hunger" (ix). Not only was Martel hungry for success, but he was also hungry for a secret ingredient for a novel--something original. When the idea of Life of Pi presented itself, Martel realized its unique value and pursued it. As a result, Martel was emotionally fed because he knew he had finally found a good story worth writing. Furthermore, the success that he found from Life of Pi drove away discouragement so he could finally feel emotionally satisfied with his career.