Mikethegourami(2)

Calculus II Manuscript 2

Attack of the Ghost Shrimp

Mike the gourami was swimming around in his freshwater aquatic environment with his TI-83 Plus Waterproof Edition. He was on his way to his 2:00 pm calculus class. However, on the way he was accosted by a band of ghost shrimp. Ghost shrimp are normally a quite peaceful species, but two things get them aggravated: warm water and higher level mathematics. These ghost shrimp were particularly aggravated. “You there, gourami! I demand you assist us with our calculus homework,” said the largest of the group, a two inch shrimp by the name of Fred. “Or else we’ll steal your calculator!” Apparently the shrimp were unaware that the University of Fishtanks employed students to act as tutors. It was unnecessary for them to use violence to get their homework done, but sometimes calculus has mind altering effects on people. Mike, deathly afraid for his calculator, was immediately cooperative. “What sort of assistance do you need?” “It’s problem 16, in chapter 11, section 4 of Sturgeon Anglerfish’s A First Course in Calculus: find ∫(x4 + 5x – 6)/(x4 – 16) dx.” After recovering from the shock that was brought upon him by the shrimp’s use of the pictorial representation of a mathematical operator in his speech, Mike began to think about how to best get out of the situation. The TI-83 Plus Waterproof Edition was unfortunately, not blessed with the ability to calculate integrals. It seemed as if the only

way to keep his calculator was to do the problem out by hand. “Okay,” he said. “We can view this as the integral of f(x)/g(x). You see, f(x) and g(x) are of the same degree. We want use long division to reduce this so that g(x) is of a greater degree than f(x). After we’ve done this, we can use the power of partial fractions to get at a point where the function is fairly simple to integrate.” So they set to it. When they divided x4 + 5x – 6 by x4 – 6 they got 1 + (5x + 10)/(x4 + 16). They then set aside the 1, and started working on how to get (5x + 10)/(x4 + 16) into something a little more integration friendly. They saw that 5x + 10 could be simplified into 5(x + 2), and x4 + 16 could be simplified out into (x + 2)(x – 2)(x2 + 4). The x + 2 cancelled out, leaving them with 5/((x – 2)(x2 + 4)). “This,” Mike said, “is where we start using partial fractions. We want to get this so that we have two fractions out of this: something over x – 2 and something over x4 + 4. You’ll notice that x4 + 4 is a polynomial. Because of this, we want a linear expression in its numerator. So we want three constants so that (c1 + c2x)(x – 2) + c3(x2 + 4) = 5.” Multiplying out this they got c1x – 2c1 + c2x2 – 2c2x + c3x2 +c34. Then they grouped these by powers of x, i.e. (c2 + c3)x2 + (c1 – 2c2)x – 2c1 + 4c3. “This is all equal to 5,” said Mike. “Therefore, we want constants such that (c2 + c3)x2 = 0, (c1 – 2c2)x = 0, and –2c1 + 4c3 = 5. We can get this by substituting the constants out for each other.” By some algebraic manipulations they discovered that c2 was equal to –c3 and vice versa, and c1 was equal to -2c3, and c3 was equal to 5/8. Plugging this back into everything else, they found that c2 was equal to -5/8, and c1 was equal to -5/4. So their two fractions were (-5x/8 – 5/4)/(x2 + 4) and (5/8)/(x – 2). They knew that that they

could evaluate these, along with 1, separately, as the sum of the derivatives is equal to the derivative of a sum. ∫1 was the easiest to evaluate; it was simply x. Pulling the constant out of (5/8)/(x – 2) gave them 5/8∫(x – 2)-1, which integrated into 5/8log(x – 2). Then they took -5/8 out of (-5x/8 – 5/4)/(x2 + 4), giving them -5/8∫(x + 2)/(x2 + 4). Integrating this out was a long and arduous process, but they finally prevailed and got 5/16tan-1(1 – x) + 5/64*log(x2 – 2x + 2) + 5/64*log(x2 + 2x + 2). So all together, it was x + 5/8log(x – 2) + 5/16tan-1(1 – x) + 5/64*log(x2 – 2x + 2) + 5/64*log(x2 + 2x + 2). “By golly, Mister Gourami, you’re awfully good at calculus,” said the ghost shrimp named Fred. “Will you help us with it again?” “Uh… sure, so long as you don’t threaten to steal my TI-83 Plus Underwater Edition calculator again,” responded Mike. So they went their separate ways, the ghost shrimp with their homework completed, and Mike to his calculus class. He had spent so much time on the problem that he was half an hour late, but as it turned out, his calculus teacher had been delayed by a warm current, so class had been cancelled. So it all turned out well for Mike in the end.