Onboard Note

Recall that indefinite integral follows 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.  For the given  integral problem: int 2x^3 cos(x^2) dx , we may apply apply u-substitution by letting:  u = x^2 then du =2x dx . Note that x^3 =x^2 *x  then 2x^3 dx = 2*x^2 *x dx or x^2 * 2x dx The integral becomes: int 2x^3 cos(x^2) dx =int x^2 *cos(x^2) *2x dx                                    = int u cos(u) du Apply formula of integration by parts: int f*g'=f*g - int g*f' . Let: f =u then f' =du        g' =cos(u) du then g=sin(u) Note: From the table of integrals, we have int cos(x) dx =sin(x) +C . int u *cos(u) du = u*sin(u) -int sin(u) du                             = usin(u) -(-cos(u)) +C                             = usin(u) + cos(u)+C Plug-in u = x^2 on  usin(u) + cos(u)+C , we get the complete indefinite integral as: int 2x^3 cos(x^2) dx =x^2sin(x^2) +cos(x^2) +C  

If a theme can be expressed in a single word, then the appropriate word for Longfellow's iconic American poem is "patriotism." Longfellow's speaker relates a story of an American patriot of the Revolution, Paul Revere, waiting for a signal as to the anticipated movements of British troops and sailors massed in and around Boston in April of 1775. After Revere receives the "intel," he sets out on a breakneck ride through the countryside surrounding Boston to spread the word to his fellow patriots as a call to arms. Though Longfellow's poem very loosely plays with the facts of that evening, the spirit of the poem captures the cooperation, diligence, courage, and determination of the men who dedicated their lives to the cause of liberty. Longfellow's speaker invokes the attention of "children" in the first stanza to inform them that he will tell the story of Paul Revere's exploits since "hardly a man is now alive/who remembers that fam...

The young man is thrown into prison for falling in love with the princess, the much beloved daughter of the king. The young man is brave and handsome, but lowborn. The princess falls in love with him without the approval—or even knowledge—of her father. In fact, the romance goes on undiscovered, we are told, for many months. When the princess's father, who very much likes to have things his own way, realizes what has happened, he has the young man cast into prison. As the text states: Never before had such a case occurred; never before had a subject dared to love the daughter of the king. We see from this action that the king is very possessive of his daughter. He also has a strong sense of social class, which means he is offended that the young man has overstepped his boundaries and dared to love a person so far above him in rank.

The Iron Curtain greatly impacted the geographic, political, and economic landscape of Europe starting in 1945 and extending to the modern day. For one, the Iron Curtain physically separated people from one another, often within their own countries. This had a devastating effect on families and the economies of Eastern Europe. These negative effects have extended into the modern day for many former Eastern Bloc nations (like Ukraine and Romania), while others have rebounded quite well. Secondly, as the Soviet Union crumbled under Gorbachev, the revolutions of 1989 sprang up, causing demands for democracy and tidal waves of courageous defiance across Eastern Europe. These political revolutions—and the new governments put into place as a result—have continued to influence global current events. As the previous answer noted, the Iron Curtain (a term coined by Winston Churchill in a famous speech) was a nonphysical boundary dividing Europe into two portions—the capitalist West and the comm...

Florence tells the story of how her great-grandparents walked all the way from Louisana to Colorado in 1859. They were both freed slaves who settled alongside the Gunnison River. She comments that her ancestors became the first black family in the entire county. Her father called them "seedfolks" because they were the first black family to settle in that location. Florence then calls the community gardeners on Gibb Street "seedfolks" because they were the first to plant their seeds in the run-down lot. Throughout the novel, Paul Fleischman tells stories about various people who immigrated to Cleveland and began planting their seeds in the garden on Gibb Street. Many of the characters throughout the novel were the first members of their families to immigrate to America, similar to how Florence's family was the first to inhabit that particular area of Colorado. Seedfolks are people in their families to move from their ancestral home and start their lives in a new ...

Pope was engaged in a literary spat with Joseph Addison—eighteenth century man of letters and editor of the hugely influential periodical The Spectator. Pope had been engaged in a major new translation of Homer's Odyssey. At the same time, a rival translation was being produced which was patronized by Addison. Not surprisingly, Addison openly expressed his opinion that it was the better of the two translations. The ever-proud Pope, never slow to take offense, went on the attack, using his biting satirical wit to answer back in "Atticus," which was later worked up into the much longer "Epistle to Dr. Arbuthnot." Pope uses the poem to settle a number of personal scores. Literary feuds were rife in eighteenth century England, and Pope was personally involved in a fair few of them. In the "Epistle" the relevant lines of the poem compare Addison to Atticus, a correspondent of the great Roman statesman Cicero. Pope uses the comparison as Atticus was someone ...

Although Henrietta Lacks was born in Roanoke, Virginia, she moved to Clover in Fairfax County at a young age after her mother passed away. Her father couldn't cope with raising his large family, so he sent Henrietta and her siblings to Clover, where they were farmed out among relatives. Life in Clover was hard for Henrietta. In common with many African Americans in rural Virginia, she worked on the tobacco plantations. Working on the plantations forced Henrietta to grow up very quickly, especially after she became a teenage mother. She subsequently married the father of her children, Day, and not long afterwards, a cousin persuaded Henrietta and her husband to move to Maryland for a better life. The state was home to a large steel-mill, the Bethlehem Steel Plant in Sparrow's Point, and Day would be able to find work there.