The 1950s and 60s are often considered polar opposites, but were the 1950s as placid and conflict-free as they seemed? Also, why did the 1960s begin on such an optimistic note yet end rather darkly?

The 1950s were not as calm as portrayed in popular culture, as this period marked the beginning of the Civil Rights Movement. This was the decade where the jailing of Rosa Parks launched the Montgomery Bus Boycott. This was also when Eisenhower sent in the National Guard, in order to protect the "Little Rock Nine" and thus desegregate the schools and uphold the Brown v. Board Supreme Court ruling.
The decade also saw extreme paranoia situated as it was in the McCarthyism era. Many were accused of being Communist, simply because they backed leftist agendas during the 1930s. Many actors and screenwriters were punished because they would not incriminate others. The period was also bad for women, as they had lost much of the autonomy they had gained during WWII and were now expected to go back to low-paying work and homemaking.
The 1960s began on a relatively positive note. The United States had a young dynamic president who promised to put a man on the moon by the end of the decade. He also launched the Peace Corps, which promised to both spread American ideals and make the world a better place. Kennedy also rose above the Southern Democratic party base and helped to establish civil rights, though his successor, Lyndon Johnson, would do more of this upon Kennedy's assassination. Johnson even declared war on poverty in his Great Society platform. The decade ended darkly, as Vietnam, a war which had gradually escalated from advisers to ground troops, turned into America's nightmare. The "living room war" showed American casualties and not enough winning of key battles. It also showed thousands of defenseless Vietnamese being put in inhumane conditions. Martin Luther King Jr. was shot in Memphis in 1968, touching off race riots all over the country. The 1968 Democratic National Convention was picketed by protesters chanting "The Whole World is Watching!" as police beat them away from the convention. Richard Nixon won the election by promising "Peace with Honor" in Vietnam, but he actually escalated the war by invading Laos and Cambodia. The United States did land a man on the moon in 1969; this was one of the bright spots of that year. All of this made the 1960s a decade which started with such promise but ended with more questions unanswered.

int tan^5 (x/2) dx Find the indefinite integral

Indefinite integrals are written in the form of int f(x) dx = F(x) +C
 where: f(x) as the integrand
           F(x) as the anti-derivative function 
          C  as the arbitrary constant known as constant of integration
To evaluate the given integral problem int tan^5(x/2) dx , we may apply u-substitution by letting: u = x/2 then du =1/2 dx or  2du= dx .
The integral becomes:
int tan^5(x/2) dx =int tan^5(u)* 2 du
Apply the basic properties of integration: int c*f(x) dx= c int f(x) dx .
int tan^5(u)* 2 du =2 int tan^5(u)du
Apply integration formula for tangent function:  int tan^n(x)dx = (tan^(n-1)(x))/(n-1)- int tan^(n-2)(x)dx .
2 int tan^5(u)du= 2 *[(tan^(5-1)(u))/(5-1)- int tan^(5-2)(u)du]
                        = 2*[(tan^(4)(u))/(4)- int tan^(3)(u)du]
Apply another set integration formula for tangent function on  int tan^(3)(u)du .
int tan^(3)(u)du = (tan^(3-1)(u))/(3-1)- int tan^(3-2)(u)du
                         = (tan^(2)(u))/(2)- int tan^(1)(u)du
                         =(tan^(2)(u))/(2)-ln (sec(u))+C
Applying  int tan^(3)(u)du =(tan^(2)(u))/(2)-ln (sec(u))+C , we get:
2 int tan^5(u)du=2*[(tan^(4)(u))/(4)- int tan^(3)(u)du]
                        =2*[(tan^(4)(u))/(4)- [(tan^(2)(u))/(2)-ln (sec(u))]]+C
                        =2*[(tan^(4)(u))/(4)-(tan^(2)(u))/(2)+ln (sec(u))]+C
                       =(tan^(4)(u))/2-tan^(2)(u)+2ln (sec(u))+C
Plug-in u = x/2 on (tan^(4)(u))/2-tan^(2)(u)+2ln (sec(u))+C , we get the indefinite integral as:
int tan^5(x/2) dx=(tan^(4)(x/2))/2-tan^(2)(x/2)+2ln (sec(x/2))+C

What allowed the Europeans to explore further into Africa and eventually conquer more?

During the nineteenth century, Europeans were able to advance into the interior of Africa. Improved military, communications, and transportation technology allowed them to do so, and more effective drugs to fight malaria helped as well.
The specific “tools of imperialism” that facilitated Europe’s colonial conquests in Africa included breech-loading repeating rifles, light field artillery, smokeless gunpowder, iron-hulled steamships, railroads, quinine, and, later, machine guns. The introduction of submarine telegraph cables enabled European colonial authorities to communicate by telegraph; this allowed them to use their limited resources in a more effective way and to send reinforcements where they were badly needed.
Quick-firing British and French rifles and artillery were important in helping British and French forces to conquer parts of Western Africa in the late nineteenth century, and the new Maxim machine guns played a decisive role in the defeat of the large-scale Sudanese Mahdi uprising against the British in the battle of Omdurman (1898). Many thousands of rebels died, but the British and Egyptian forces lost just a few dozen of their own people. The Maxim machine guns devastated the courageous Ndebele warriors in South Africa (1893), thus defeating one of the strongest states that opposed the British colonial advance.
Where European troops did not have an overwhelming technological advantage and African forces were reasonably well-trained in the use of modern weapons, the European invading armies sometimes experienced crushing defeats. The Ethiopian victory over Italian forces in the battle of Adwa (1896) is a notable example of this.

Who is Raymond Sintes and what is the word around the neighborhood regarding Raymond?

Raymond Sintes is a foil to Meursault. This means that he provides a contrast to the protagonist, emphasizing certain characteristics of his. If Meursault is amoral, his neighbor Sintes is completely immoral. He's a pretty unpleasant character, to put it mildly. He is a macho chauvinist who physically abuses his mistress and generally treats her like dirt. It is Sintes's contempt for his mistress that leads to Meursault becoming embroiled in a scheme that ultimately ends in murder. Sintes is devious and manipulative and thus is able to get Meursault to do his bidding.
The reputation that Sintes has in the neighborhood is that of a pimp. And he certainly gives the impression of being one, with his flashy clothes and his reputation of violence towards women. Sintes is indeed subsequently exposed as a pimp in court, but he remains somewhat cagey about what he does for a living. Despite everything, Sintes still seems keen to maintain a certain reputation in the neighborhood.

During the 1950s, who did advertisers begin to target?

Television changed advertising in the 1950s.  Millions of American families watched TV and the commercials that ran.  Popular celebrities starred in commercial ads showing new products.  Ads began to target children and teenagers.  Ads targeted toward children showed new and exciting toys.  Ads for teenagers showed everything from phonographs to sodas to clothes.
The 1950s were a time of increased consumerism.  The previous two decades had been filled with hardships.  The Great Depression of the 1930s and World War II in the 1940s had forced consumers to cut back.  The economy was flourishing by the 1950s, and American consumption reflected this.  It was in the 1950s that advertisers began researching their demographics.  They wanted ads to target specific demographics.  Besides children and teenagers, women were frequently targeted.  Many women were housewives, so ads for cleaning products were common.  Smoking was a common habit for adults, and cigarette ads dominated the television screens during Prime Time.

What ailment might the narrator suffer? How does the narrator use personification throughout the story?

It seems that the narrator is suffering from what we would call postpartum depression, though they did not have such a diagnosis when Gilman wrote the story. The narrator's husband, a doctor, tells friends and family that she has only a "temporary nervous depression—a slight hysterical tendency." A diagnosis of hysteria tended to function as a sort of catchall for any mental problems that seemed particularly associated with women: in fact, the word hysteria comes from the Greek word for uterus. Further, the narrator talks about how glad she is that "Mary is so good with the baby. Such a dear baby! And yet I cannot be with him, it makes me so nervous." Her confession that being with her baby makes her feel nervous is a good indicator that the state of her mental health has a great deal to do with the fact that she's recently given birth and that she is not, perhaps, connecting with her child in the way that one might hope or expect.
Throughout the story, the narrator routinely personifies the wallpaper in her room. For example, she says,

There is a recurrent spot where the pattern lolls like a broken neck and two bulbous eyes stare at you upside down [....]. Up and down and sideways they crawl, and those absurd, unblinking eyes are everywhere [....]. I never saw so much expression in an inanimate thing before, and we all know how much expression they have! I used to lie awake as a child and get more entertainment and terror out of blank walls and plain furniture than most children could find in a toy-store. I remember what a kindly wink the knobs of our big, old bureau used to have, and there was one chair that always seemed like a strong friend.

She gives the design of the wallpaper a neck and eyes and the ability to crawl. Not only does she attribute life to the wallpaper itself, but it also seems to have been a pattern with her, throughout her life, to personify all kinds of objects. She talks about how inanimate things always seem to have a great deal of expression to her: a pretty atypical description of things like walls and wardrobes. This helps gives us some insight into her imagination as well; she is clearly very creative and thoughtful and that must make it all the more frustrating and upsetting for her that her husband has removed all means of mental stimulation from her room.

Can you help me with writing a descriptive essay on the experience of a classroom without a teacher?

Since you are tasked with describing something you must imagine, your first step is to close your eyes and visualize a teacher-less classroom. What does that look like in your mind's eye? Jot short notes or phrases that will help you remember what you're seeing. The more clearly you visualize, the better your essay will be. Imagine you are a student in this classroom. What does it feel like? How do you like it? Note any feelings or sensations associated with your image.
Using your imagined picture, choose a thesis statement. It might be something like, "A classroom without a teacher is total chaos," or "Students learn better without a teacher." Let your image lead you. Then, organize your essay around your thesis. A good hook might be to describe your classroom to the reader; try to make readers see your classroom the same way you do. Throughout the essay, keep picturing your teacher-less classroom. Use that image to support your thesis.
For example, in your imagination are the students engaged and learning? How can you tell? What are they doing? Their actions and expressions are what you want to describe. Why do students learn better without a teacher? Because you "saw" them working individually at computers, and helping each other with questions. You "saw" how much more efficient it was. Or maybe you pictured unruly kids talking, throwing things, and running around the class. Describing this scene to your readers supports a thesis that a classroom without a teacher is total chaos.
It is said a picture is worth 1,000 words. Create a clear picture in your mind, and you'll have no trouble writing a few-hundred word essay. Have fun with it! And remember there is no right or wrong description of what this new type of classroom would be like. Your experience is unique to you.

A descriptive essay is an essay topic that English or writing teachers like to frequently give at some point in the year. At first look, the process can be daunting, but the writing prompt given is fairly specific, so that will help a great deal. I'm not allowed to write the essay for you; however, it does seem like a potentially really fun topic. The descriptive essay itself is an essay that asks a writer to describe something. It could be an object, place, person, experience, emotion, or situation. There is a lot of flexibility within the form of this essay, but the main goal is that the piece paints a vivid image in the mind of the reader. With that in mind, I'd like to offer some tips to get you going in the right direction.
One of the first things to decide is what the topic is going to be. This has been decided for you, but that's only a small part of the rhetorical situation. Another thing that has to be decided is who is narrating. Will you, as a student within this classroom, be the person that is narrating the story? This is where the story's flexibility comes in because your narrator doesn't have to be you or even another student within the classroom without a teacher. Your narrator could be a literal fly on the wall. You could even get creepy and have your narrator watching the events from a control room using hidden cameras and microphones throughout the room. You also have to decide on what genre you are going to write. Will your essay be a comedy, a horror story, a news piece, or something else? Finally, another thing to think about is where your essay will take place. What grade level doesn't have a teacher? A first grade classroom without a teacher is going to be a very different place than a college classroom.
The next thing to do is create an attention grabbing hook. You need to get your reader's attention right away. The first sentence is key. You could start by asking your reader a question in order to get them thinking about the situation. You could start with a quote. This could be interesting to play around with because the quote could be completely made up. It could be a piece of dialogue that a fictional student shouts out at the top of his/her lungs about the teacher being gone. The hook could be a bold, declarative sentence that sets the reader on edge. This worked amazingly well for Andy Weir's book The Martian.
Now that the first sentence is done, you can begin writing about the experience that a classroom without a teacher is to your narrator. An important thing to remember throughout the descriptive essay is to describe what the situation is like. Use vivid language to really help the reader understand what is it like in the classroom. Use similes and metaphors more than you normally would for a school essay. If you are drawing a blank on how to describe something or what to actually describe, consider your five senses. Tell your reader what it is that you see, smell, hear, feel, and even taste. For example, "It was five minutes into the class period, and the teacher was still absent. The students were all starting to get the same feeling. They could practically taste the freedom that was slowly wafting into the classroom like the smell of freshly fried bacon."
A final recommendation is to stay organized. You might be tempted to write this essay in a stream of consciousness style. That could definitely work, but I don't recommend it. I understand that writing it in stream of consciousness makes sense, but it makes sense to you. Your reader is not you, and his/her stream flows slightly differently. The stream of consciousness will likely be a very jarring read for your reader, so stick with an organized format. Perhaps write each paragraph about what each of your senses are experiencing. This organization isn't perhaps realistic, but it does allow you to guide your reader into reading and experiencing each item in full detail in the order that you want the reader to experience it in.
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