Onboard Note

int(8x)/(x^3+x^2-x-1)dx (8x)/(x^3+x^2-x-1)=(8x)/((x^3+x^2)-1(x+1)) =(8x)/((x^2(x+1)-1(x+1))) =(8x)/((x+1)(x^2-1)) =(8x)/((x+1)(x+1)(x-1)) =(8x)/((x-1)(x+1)^2) Now let's form the partial fraction template, (8x)/((x-1)(x+1)^2)=A/(x-1)+B/(x+1)+C/(x+1)^2 Multiply the equation by the denominator, 8x=A(x+1)^2+B(x-1)(x+1)+C(x-1) 8x=A(x^2+2x+1)+B(x^2-1)+C(x-1) 8x=Ax^2+2Ax+A+Bx^2-B+Cx-C 8x=(A+B)x^2+(2A+C)x+A-B-C Comparing the coefficients of the like terms, A+B=0 -----------------(1) 2A+C=8 -----------------(2) A-B-C=0 ---------------(3) From equation 1, B=-A Substitute B in equation 3, A-(-A)-C=0 2A-C=0 ---------------(4) Now add equations 2 and 4, 4A=8 A=8/4 A=2 B=-A=-2 Plug in the value of A in equation 4, 2(2)-C=0 C=4 Plug in the values of A, B and C in the partial fraction template, (8x)/((x-1)(x+1)^2)=2/(x-1)+(-2)/(x+1)+4/(x+1)^2 int(8x)/(x^3+x^2-x-1)dx=int(2/(x-1)-2/(x+1)+4/(x+1)^2)dx Apply the sum rule, =int2/(x-1)dx-int2/(x+1)dx+int4/(x+1)^2dx Take the constant out, =...

When Orwell relates his experience with the elephant in “Shooting an Elephant” it gives some insight into his own psyche as well as the structure of imperialism. Perhaps the most intriguing situation that arises in the story is how the mob has more power over Orwell than he, their supposed military governor, has over them. In spite of knowing the elephant could be captured and tamed, the mob calls for its death, and the cries overcome Orwell, causing him to shoot and kill the beast. In this moment, he criticizes imperialism, showing that the leaders are controlled by the masses just as much as, if not more so than, the other way around. The insight into Orwell’s own mind is similar. He reveals a weakness and malleability in this moment. It shows that Orwell can be swayed by the opinion of the others, and that he is not immune to this idea, even if he recognizes it. "Shooting an Elephant" contains George Orwell's ruminations concerning an experience he had back when he was...

There's a particularly striking example of animal imagery in Abigail's description of John Proctor as "sweating like a stallion." She's referring of course to their illicit affair, which will cause no end of trouble to John, both with his wife Elizabeth and with Abigail herself when she sets out to destroy him out of revenge for his ending their relationship. This vivid piece of imagery perfectly conveys the sheer animal intensity of the affair between Abigail and John. It also highlights its illicit nature. In this part of the world, in the Calvinist theocracy that is Salem, men and women should not behave like this. They are supposed to be God-fearing, upright citizens: members of God's elect. They are not supposed to succumb to primal urges like animals. And yet that's precisely what John and Abigail have done. In doing so, they've separated themselves from respectable society, a process that will be accelerated in John's case when he's fals...

To simplify the given complex fraction (x/3-6)/(10+4/x) , we may look for the LCD or least common denominator. The denominators are x  and 3 . Both are distinct factors. Thus, we get the LCD by getting the product of the distinct factors from denominator side of each term. LCD =3*x=3x Multiply each term by the LCD=3x . (x/3*3x-6*3x)/(10*3x+4/x*3x) (x^2-18x)/(30x+12) Another method is to simplify top and bottom as single fraction. Let 6=18/3 and 10=(10x)/x . (x/3-6)/(10+4/x) (x/3-18/3)/((10x)/x+4/x) ((x-18)/3)/((10x+4)/x) Flip the fraction at the bottom to proceed to multiplication. ((x-18)/3)*(x/(10x+4)) Multiply across fractions. ((x-18)*x)/(3*(10x+4)) (x^2-18x)/(30x+12) The complex fraction (x/3-6)/(10+4/x) simplifies to  (x^2-18x)/(30x+12) .

Throughout the story "Indian Education," there is only one mention of the narrator's cousin, Steven Ford, but  the idea of this character is present throughout the narrative. When in fifth grade, the narrator, Victor, "picked up a basketball for the first time" and thought about "all those possibilities and angles. It was mathematics, geometry. It was beautiful." However, he quickly contrasts his decision to dedicate himself to basketball to his cousin, who "sniffed rubber cement from a paper bag and leaned back on the merry-go-round. His ears rang, his mouth was dry and everyone seemed so far away." Throughout the rest of the story, Victor attempts to make something of himself through basketball. He goes to the white high school and is a star on the basketball team. The reader is left assuming that Steven Ford was still on that symbolic merry-go-round getting high with "that buzz in his head, all those colors and noises." But Victo...

In Brave New World social class is based on a rigid hierarchy created by genetic engineering. The state planners have designed levels of intelligence in the population which are a futuristic counterpart to the class-structure of the past world. Each level is named by a letter of the Greek alphabet—alpha through epsilon. Within each stratum there are sub-categories, such as minus, plus, and double-plus. In the story, for instance, Bernard is an Alpha (more specifically an Alpha-plus), a member of the most intelligent class, and Lenina is a Beta, the second highest main level. There are references through the story to the "lowest" level of the hierarchy, the Epsilons, who are evidently mentally-challenged people. As is typical of dystopian novels, in Brave New World there are individuals or segments of the population who represent "the past" and have remained outside the state-controlled modernized world. Such people are encountered on the "Savage Reservation...

Below are the graphs of the velocity functions of two particles, where $t$ is measured in seconds. At what time each particle speeding up? Slowing down? Explain. a.) The particle is speeding up when the velocity is increasing (either in the positive or negative direction). On the other hand the particle is slowing down when the velocity is decreasing. Based from the graph we can say that the particle is speeding up on intervals $0 \leq t \leq 1$ and $2 \leq t \leq 3$ (speeding up on the negative direction). While the particle is slowing down on interval $1 b.) Based from the graph, we can say that the particle is speeding up at intervals $1