Think of the number zero as the center of the number line. It is neither positive nor negative. Negative numbers are on the left side of zero on the number line, which stretches to infinity going left. Positive numbers are on the right side of zero on the number line, which stretches to infinity going right.

Figure 1: The number line.

The part of the number line that begins at zero and stretches to infinity going to the right consists of all numbers that are more than zero. In math symbol language the symbol for greater than or equal to zero is ≥, so quick shorthand for positive numbers is N ≥ 0.

It took many hundreds of years for people to start using negative numbers. Some philosophers and mathematicians were against them until almost the 18^{th} century. People think of them as opposites to positive numbers. For example, a temperature of -40^{o} C (-40^{o} F) is opposite of a temperature of 40^{o} C (104^{o} F). Mt. Everest is over 8839 m (29000 feet) above sea level. In the opposite direction, the distance to the Challenger Deep in the Pacific Ocean is over -10972 m (-36000 feet). The negative number shows that the distance is below sea level. If your bank account has a balance of $100 in it, that is different than having a balance of -$100 (overdrawn). In math shorthand language, for negative numbers is N≤ 0, less than or equal to zero. If the time until a rocket launch is T – 30 minutes, it is still on the launch pad. If the time is T 30 minutes it has left the launch pad.

Figure 2: This thermometer shows both positive and negative temperatures in Celsius and Fahrenheit.

If two positive numbers are added, the result is positive. Suppose that 492 and 3 are the positive numbers. Their sum, 495, is also a positive number. If two negative numbers are added, the result is negative. If -3.2 and -10 are added, their sum -13.2 is negative. If a positive number is added to a negative number and the absolute value of the positive number is larger, the answer will be another positive number, such as 5 and -2. The sum is 3. However, if -8 and 3 are added, the sum is -5. In terms of the number line, -8 is further to the left than 3 is to the right. Speaking math language, if a positive number is added to a negative number and the absolute value of the negative number is larger, the sum will be negative.

Figure 3: Addition of positive and negative numbers.

If two positive numbers are multiplied, the result is always positive. Similarly, if two negative numbers are multiplied, the result is positive. Another way to say this is the same signs are always positive, so that 3∙7 = 21 and -6∙ -8 is 48. If the signs are opposite, such as 4∙-3 =-12, or -7 ∙ 8 = -56, the result is always negative.

Figure 4: Multiplication of positive and negative numbers.

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]]>A chemical reaction can be described by discussing the chemical substances at all steps. It is more efficient to use a chemical equation than words, and there are agreed-upon conventions to summarize the reaction, the substances that take part in it, the products, and the amounts.

A chemical reaction starts with substances (often elements or compounds) that take part in it. They are called the reactants. Something happens to cause a change, and some part of the reactants are rearranged. Chemical bonds are formed, reformed, and broken, and new types of substances are made. Those new substances are the products of the reaction. For example, two atoms of hydrogen for every atom of oxygen combine to form water, so that 2H_{2 } + O_{2}→ 2 H_{2}O.

Figure 1: Equation for the chemical reaction between hydrogen and oxygen to produce water.

The symbols for the elements are taken from the periodic table, such as H for hydrogen, O for oxygen, Au for gold, K for potassium, and so on. The reactants are on the left of the equation, while the products are on the right. They are separated by an arrow (→) that points in the direction the reaction is taking. If the arrow is double and points in both directions (D) that means that the reactants and the products are in equilibrium. Heat is represented by the delta sign (∆) placed above the arrows. If a gas is released, it is shown by an arrow pointing upward (↑), and if a solid is precipitated, an arrow points downward (↓). If substances are added during a reaction, a plus sign is placed between the substances.

Figure 2: Methane and oxygen combine to form carbon dioxide and water.

Chemical formulas for a particular compound give a great deal of information that is subject to interpretation. The chemical formula for water is H_{2}O. That can mean that there are 2 hydrogen atoms for every oxygen atom, one molecule of water, 1 mole (abbreviated mol) of water, 1 molar mass of water, 6.022 x 10^{23} molecules of water (Avogadro’s number), or 18.02 grams of water.

When a chemical equation is balanced, it is clear what substances are the reactants, which are the products, how much of each substance is involved, as well as their relationship to each other, and the steps that occur during the reaction. Suppose that hydrogen gas reacts with chlorine gas to produce hydrogen chloride, which is also a gas. In symbol form, H_{2 }(*g*) + Cl_{2}(*g*) → 2 HCl (*g*).

Figure 3: When a chemical equation is balanced, it has the right amount of reactants and products on both sides.

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]]>Many students in high school decide to take a chemistry class to fulfil a science requirement for graduation, in order to attend college, or as a basis for a later career in medicine or one of the other sciences. In order to be successful in chemistry, students take part in lab experiments and participate in class discussions, students should be able to read assigned material carefully, apply the vocabulary of chemistry, and use mathematics appropriately when needed.

Figure 1: Success in chemistry class depends on four elements.

For some students, the best part of chemistry class is doing lab experiments. They learn to follow directions precisely, and keep careful, detailed records in their chemistry notebooks. For others, the worst part of chemistry class is doing lab experiments, because something always goes wrong. They also learn to keep detailed records of what happened. Sometimes the teacher does a demonstration of a chemical reaction as a springboard for class discussion. In many chemistry classes, there is a class participation requirement, and teachers take it seriously.

Figure 2: Students and teachers participate in chemistry classes.

The history of chemistry is exciting, with experiments, discoveries, a lot of twists and turns, and visions of the future. Unfortunately, the assigned textbook may or may not convey the same sense of anticipation. There are a lot of facts to memorize, chemical formulas, lists, and terms with unfamiliar meanings. When a textbook chapter is assigned, first do some prereading. Note the title of the chapter, and the topics the chapter will cover? Those topics are given as topic subheadings. Are there diagrams of processes and equations? Make notes of those things before reading the chapter, and then read the chapter. This will give a study outline as well as organizing thoughts, and can be used as a study guide before the test.

The language of chemistry is specialized. It includes the arrangement of elements in the periodic table, with their abbreviations; the symbol language of chemical equations and formulas; all kinds of terminology for chemical processes; the laws of chemistry; and rigorous application of the scientific method. Some of the language will come from the reading, and some will come from the class itself.

Figure 3: The specialized vocabulary of chemistry includes the arrangement and elements of the periodic table.

The sciences are filled with applications of mathematics, and chemistry is no exception. Many high schools require students to have taken algebra and geometry before taking chemistry, or be taking a math class at the same time. In order to be successful in balancing chemical equations, students are required to use those mathematical concepts, which enables them to learn how equations actually work.

Figure 4: An example of the application of mathematics to chemical reactions.

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]]>Accurate observations of phenomena are one of the keys to success in lab sciences such as chemistry, physics, and biology. The metric system, or international system (SI), is used to describe measurements of many quantities, such as length, mass, volume, and temperature.

The measurement of length was one of the first measurements to be standardized, because the measurement of length varied between country to country and sometimes throughout periods of time. In ancient times, the measurement of length might be defined as the length of a king’s stride. Suppose one king was tall and had a long stride and the king in the next kingdom over was shorter and had a short stride. Since communication between scientists is often international, scientists had to agree on one definition. As measurements have become more precise, the definition of the meter has become more precise. A meter in conventional units is 39.37 inches, a little more than a yard. The meter has been defined in the scientific community as the distance that light travels in a vacuum during 1/299,792,458 of a second. Using the metric system, a centimeter is 1/100 of a meter (2.54 inches). Many observations made in science class use centimeters.

Figure 2: Methods for measuring length.

Mass is measured using a balance scale, which is often crucial while conducting chemical experiments, while weight is often measured on a spring-type scale. (Think of the scale in the bathroom or in the doctor’s office.) In the metric system, mass is measured by the kilogram (kg), in conventional units 2.2 pounds. A gram, which is 1/1000 of a kilogram, is 0.035 of an ounce in conventional units, so it is too small for many uses.

Figure 3: A balance scale used in scientific measurement of mass.

Volume, as measured in chemistry, is the amount of space that matter occupies. It is most often measured by the liter (L), 1.057 qt. in conventional units, or the milliliter (mL), 1/1000 of a liter, about 0.0338 of an ounce. It is often measured by cylinders, flasks, pipettes, or syringes in and out of the laboratory. If you have gone to the doctor and had a shot or other liquid medication, the proper dosage is measured in milliliters rather than ounces.

Figure 4: Precise measurement ensures the proper dosage.

Figure 5: Volumetric flasks measure liquid chemicals accurately.

Temperature, as measured in science, is a measure of how intense the heat is in an object. It is usually measured in Celsius or Kelvin. The Kelvin scale is used to measure very cold temperatures, as 0 Kelvin is absolute zero, as well as very hot temperatures, such as temperatures in the sun and other stars. The degrees Celsius are written with the degree sign (^{o}) and the letter C, while the degrees Kelvin use the letter K without the degree sign. In conventional measurement, the boiling point of water is 212^{o} F. In Celsius, the boiling point of water is 100^{o} C; in Kelvin, 373.

Figure 6: Comparison of Celsius and Kelvin temperature scales.

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]]>Mass and weight define distinct but related concepts. Mass refers to the quantity of matter an object contains, is measured by a balance, and is invariant no matter where it is located. In contrast, weight depends on the pull of gravity on the object, is measured on a scale, and really depends on where the object is located.

In science, the mass of an object is often more relevant than its weight. Suppose one has a kilogram of cotton candy and a kilogram of gold. They would both weigh the same (1000 g), but the volume of space the kilogram of cotton candy would take up would be very large, sweet, and sickening, made of spun sugar and air. In contrast, a kilogram of gold would take up much less volume, and be much more valuable at the current rate of prices.

Figure 1: Balancing cotton candy and gold.

Mass refers to the quantity of matter an object contains, while weight depends on the pull of gravity on an object. The mass doesn’t change, but its weight does. Most of the time this doesn’t matter on earth, but it can in scientific measurements. Suppose an engineer, Dr. Xenia Xavier, in Research and Development is testing the effects of mechanical stress on a new metal alloy. She devises an experiment before using the alloy Aluminex in the jet engine, before going to the expense of having machinists fabricate Aluminex parts. Several ingots are made of Aluminex, and each ingot has a mass of one kilogram. In every condition of the experiment, the ingot is measured and weighed. The weight of the ingot is one kilogram when the jet is sitting on the runway. That ingot is taken out and replaced with another. In the next condition, the weight of the ingot is measured at 5kg while the jet is climbing. Tests are done on the metal to see how it performs under that weight. The ingot is replaced with another for the third condition, where the jet climbs to enter free flight. The 1-kg mass now weighs nothing.

Figure 2: Mass is a measure of the quantity of matter, while weight is a measure of the pull of gravity.

In the previous set of experiments, a known mass was compared under different conditions where its weight changed. Suppose the tests of Aluminex have gone far enough that pieces of Aluminex have been machined into parts. Now that the ingots have been shaped into parts, the mass of each part is unknown. Another member of the R and D team, a chemist, Dr. Ying Yee, measures the mass of each part using a balance. He puts the metal part in one pan of the balance and adds the exact amount of weights of known mass to the other pan. After the mass of a part is measured, he measures its weight using a spring-type scale.

Figure 3: Scientists use special equipment to measure mass and weight with precision.

Now for the last situation comparing mass and weight. Another member of the R and D team is going to test Aluminex, a computer specialist named Dr. Zelda Zygonowitz. Will Aluminex be a good metal to use in the rockets that the company has under development? Dr. X measured Aluminex using the SST to show that the weight of a known mass varied at different points in the flight, when the gravity changed, and Dr. Y measured the mass and weight of parts made of Aluminex. Dr. Z does many of her experiments using a computer, because the expense of traveling into space is too great. She performs calculations showing although the mass of Dr. Y’s parts do not change, the weight of that same part is different on the moon, where gravity is lighter, on Jupiter where the gravity is heavier, and near the event horizon of a black hole, where gravity is much heavier. It remains for Dr. Z’s descendant in the 24^{th} century, an astrochemist named Dr. Zena Zane, to test those parameters on a five-year deep space mission.

Figure 4: Presenting test results during a deep space mission.

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]]>Math is an integral part of life, yet some students find it more than difficult to study it because of anxiety over doing math. There are practical solutions that will lead from math avoidance to math success, but that will not come overnight. The coiled fire-breathing dragon named Math does not magically turn into a tiny lizard without some hard work.

Figure 1: Shrinking a dragon takes strategy.

Perhaps your math anxiety stems from a second-grade teacher who believed that “girls can’t do math” or a seventh-grade teacher who taught that some students were naturally better at math, and you weren’t one of them. Whatever the cause, the idea of mathematics and thinking with numbers sets up a cycle of paranoia and panic.

You, the student, may have to take algebra as a graduation requirement, or prepare for the math portion of the SAT, or the math portion of any of the standardized tests in your state to fulfill graduation requirements. When you are out of school, you will still need math: figuring out a budget, keeping a checkbook balanced, or paying for groceries. Maybe you work part-time at a fast-food restaurant and work with numbers all the time with the help of a cash register. Your future may include working in an office, calculating financial eligibility for clients, or calculating the cost of parts and labor as an auto mechanic. Numbers are all around us, whether we like it… or not.

Figure 2: Numbers are all around us. Do you see the parabola or feel like cliff-diving?

You are sitting in math class, and the teacher says something that you don’t understand. What do you do? You have three choices: Raise your hand and ask the question now, write it down (I don’t get X), and ask later, or hope that subject isn’t on the test. Fast-forward to Future You sitting in a class that involves math. (Scary, huh?) The teacher says something in math language, and they might as well be teaching the class in Old High Vulcan. “WAIT A MINUTE,” Future You says inside. “I DON’T GET THIS.” At the same time, your classmate Dave says out loud, “Go back and explain that one again, Teach.” He didn’t get it either. The teacher explains that concept, and then adds another concept. This time your classmate Cindy Lou says, “I don’t understand. Can you try that one a different way?” You get the picture. Everyone doesn’t understand sometimes. Ask your teacher questions, or your math tutor, or the textbook when you are working homework. (The family dog may be confused, but Fido will get over it.)

Figure 3: Choose your strategy when you don’t understand something in class.

If you play an instrument, you’ve heard this one. You get to Carnegie Hall by practicing every day. Your music teacher expects you to play scales or vocalize over and over again. If you are on a sports team, your coach has you do drills before the big game, running the same track day after day. If you drive a car, you practice driving before you get your driver’s license. Math is no different. Think of your math homework as math practice, and your tutor as your coach. Some of the strategies can include ways to estimate the answer before working on the problem, breaking the problem into parts to solve simpler problems, sketching the problem you are solving before writing the equation, and using what you know.

Figure 5: Think of your math tutor as your coach.

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]]>Asking questions inside and outside of math class and using your teacher and math tutor as your math coach are the first steps in slaying the dragon of anxiety over math. Some other strategies to add to your math toolbox include using what you already know, breaking the problem into parts to solve simpler problems, sketching the problem before writing the equation, and estimating the answer before solving the problem.

Every written language has an alphabet that represents sounds that make up words, words that make up sentences, and rules for generating them. Math has a language all its own, with numbers and concepts that are represented by symbols. Those symbols are combined with other symbols to form equations and inequalities. Math is also sequential, and concepts build upon other concepts. This is a big advantage to a student, because a lot of the time spent in math class is review. That means that you as a student can find the concepts you already understand and build upon them. (A psychologist named Vygotsky calls that using scaffolding). Suppose you know that variables can represent numbers in an equation, but need help getting to the next step of solving the equation. You know that 4 + y = 10 means that y represents a number. The next step is using the additive inverse on both sides of the equation. In other words, 4 – 4 + y = 10 – 4, or y = 6.

Figure 1: Apply what you know and then build upon it.

This is such a useful tool that mathematicians use it all the time, even when things get really difficult. (In fact, it is a good strategy to use in everyday life.) Any huge problem can be broken up into smaller parts, which are solvable, then those smaller parts can be combined. A recent Calculus post on using the chain rule showed how this could be done for a complex problem involving functions and derivatives. It’s also another way to use what you know.

Figure 2: Break the problem into parts, solve the simpler parts, and then put the parts together.

This tool is a way to draw a picture, write a list, or sketch a problem in a way that makes sense to you BEFORE diving in to the equation. It’s also a way to make sure you are solving the same problem as is being asked. Then you can use what you know to translate the sketch into the equation.

Figure 3: Sketch the problem before solving the equation.

This is a tool that can be applied to problems in a textbook or problems in everyday life. It is another way to see if an answer makes sense. One leg of a right triangle measures 5 cm, and the other leg of the right triangle measures 6cm. How can the hypotenuse be estimated? I know that the square of 5cm is 25cm, and the square of 6cm is 36cm, and 25 + 36 = 61, so the measure of the hypotenuse will be the square root of 61. The number 61 is not a perfect square, but it’s close to 64 which is. I can narrow it down to between 7 and 8, and that it’s probably closer to 8 than 7. That’s without a calculator. (The answer to two decimal places is 7.81).

Figure 4: Estimate the answer before finding an exact solution, then see how close the actual answer is to the estimate.

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A complex rational expression contains a fraction within a fraction. In math language, it is a rational expression that has one or more rational expressions contained within the numerator, denominator, or both. It may or may not contain variables within it. The fractions in the numerator or denominator must be simplified first, before the other rational expression can be tackled.

A rational expression is a ratio. Examples of simple ratios that do not contain variables are 2/3, 6/7, and 7/8. Rational expressions can also include variables, such as (3x)/7, (9z + 2)/4, or 3/(y + 1). They can be simplified in one or two steps. Complex rational expressions (also called complex fractional expressions) are ratios that contain fractions within fractions. Simplifying them often takes more than one or two steps.

Figure 1: A complex fraction in symbol form.

Suppose a rational expression contains the terms (2/3 + 1/4) in the numerator and 5 in the denominator. In one method, the first step in simplifying the rational expression is to find the LCM for the fractions in the numerator, changing each fraction to the equivalent common denominator, and performing the operation. In this example 2/3 becomes 8/12 and ¼ becomes 3/12. Adding 8/12 and 3/12 equals 11/12. If one or more variables in the rational expression is in the numerator, the process is similar. Suppose another complex rational expression contains a variable in the numerator, so that the terms are ([4x]/7 + x/9)/2. In that case, the first step would be similar: The LCM of 7 and 9 is 63, so the numerator becomes 36x/63 + 7x/63 or 43x/63. The denominator is still 2.

Figure 2: Simplifying the ratio when a fraction is in the numerator.

If the rational expression is in the denominator, the process is similar to solving the rational expression in the denominator. However, if there is a variable in the denominator, the denominator cannot equal zero. That would be the same thing as dividing by zero, which is undefined by definition. Suppose the numerator is 3 and the denominator is 11/12 + 1/20. Using the same method, the LCM is 60, so the fractions in the denominator become 55/60 + 3/60 or 58/60. (In turn, 58/60 can be simplified to 29/30.) The rational expression becomes 3/(29/30).

Figure 3: Simplifying the ratio when a fraction is in the denominator.

Whether the expression is (11/12)/5, (43x/63)/2, 3/(29/30), 12/(3x + 1)/2, or even (3/(2x + 1)/1/2), solving the ratio in the numerator or denominator is only part of the process. The ratio is not fully simplified until the fraction within the fraction is done. Suppose the expression is (11/12)/5. That is equal to 11/12 ∙ 1/5 = 11/60. Similarly, (43x/63)2 means the same thing as 43x/63 ∙ ½ or 43x/126.

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]]>The slope-intercept equation (y = mx + b) is a linear equation that gives both the slope of a line and where the line crosses the y-axis. It has many applications outside of mathematics, from construction and describing the grade of terrain to describing the results of scientific experiments if there is a linear relationship between the independent and dependent variable.

The slope of a linear equation is the relationship between the change in the x variable and the change in the y variable. It is a ratio, defined as the change in y (the rise) over the change in x (the run). In symbol form, the slope of a line containing the points (x_{1, }y_{1}) and (x_{2, }y_{2}) equals the ratio of (y_{2}-y_{1})/(x_{2}-x_{1}). When solving a linear equation, the y coordinates must be subtracted in the same order as the x coordinates in order to solve the equation, and describe the line correctly. Suppose a line contains the points (1, 2) and (4, 4). The slope will equal the ratio of (4-2)/ (4-1) or 2/3.

Figure 1: The slope of a line is a ratio of the change in y (the rise) over the change in x (the run).

The slope of a line also tells how it slants. Suppose that m is positive, such as in the previous example of 2/3. The line slants upwards from left to right. If m is negative, the line slants downward from left to right. The larger the slant, the greater the slope. If the line is parallel to the x-axis, the slope is 0. A line that is parallel to the y axis, however, has an undefined slope, as it is like dividing by zero.

Figure 2: If there is no change in y, the slope of the line is 0, and the line is parallel to the x axis.

The y-intercept is the point where the line crosses the y-axis. Recall in a graph of Cartesian coordinates there is an x-axis (the horizontal one) and a y-axis (the vertical one). In a linear equation, the y-intercept is a single point, a constant.

Figure 3: The y intercept is a single point, a constant

Suppose that a hill has a grade of 10%. That is the rise over the run, or for every horizontal distance of 100 feet the road rises 10 feet. Snoqualmie Pass is the largest pass in Washington State that is kept open year-round, and a major east-west route over I-90. A route from Seattle to the summit of Snoqualmie pass goes from an elevation of 520 feet to elevation of 3022 feet, a rise of approximately 2502 feet. Although the distance between Seattle and Snoqualmie pass is about 50 miles, the grade varies and frequently exceeds the recommended average grade of 6% for interstate highways. This makes it more difficult for long-distance truckers carrying heavy loads between the eastern and western parts of the state, so that highway engineers are constantly modifying the highway to improve conditions.

Figure 4: Snoqualmie Pass is a major route over I-90, and is kept open year-round as road conditions allow.

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]]>Suppose we have an integral which appears unsolvable by ordinary means, such as

∫xsin(x)dx In many cases, we can do what is called integration by parts, which is where we split the equation we are taking the integral of into two parts, with the goal of simplifying the equation such that we can reduce it down to an equation we know how to solve.

Figure 1: General formula for Integration by Parts

In essence, what this means is that we are going to attempt to split up the equation in a way such that we can eliminate one of the equations inside the integral, allowing us to end up with a form that we know how to solve.

∫xsin(x)dx

Let u(x)=x and v’(x)=sin(x)

Then we need to find u’(x) and v(x), so u’(x) = d/dx(x) = 1

Next, we need to find v(x), which we do by doing ∫v’(x)dx, or in this case, ∫sin(x)dx = -cos(x)

Now that we have u(x), v(x), u’(x), and v’(x), we can plug everything in, resulting in the following:

∫xsin(x)dx = x*(-cos(x)) – ∫1*(-cos(x))dx

= -xcos(x)+∫cos(x)dx

= -xcos(x)+sin(x)+C

In general, we will want to treat the polynomial as u(x) and the other team as v(x) where possible. In addition, Integration by Parts can be done multiple times. Suppose we have the equation

∫x^{3}e^{x}dx

Then we will let u(x) = x^{3} and v’(x) = e^{x}, then u’(x) = 3x^{2} and v(x)=e^{x}, and we find that

∫x^{3}e^{x}dx = x^{3}e^{x} – ∫3x^{2}e^{x}dx

Then we can do the same procedure with the slightly easier integral ∫3x^{2}e^{x}dx

Let u(x) = 3x^{2} and v’(x) = e^{x}, then u’(x) = 6x and v(x) = e^{x}

Then ∫x^{3}e^{x}dx = x^{3}e^{x} – ∫3x^{2}e^{x}dx = x^{3}e^{x} – (3x^{2}e^{x} – ∫6xe^{x}dx)

And then with ∫6xe^{x}dx, we split it one last time into u(x) = 6x, v’(x) = e^{x}, and therefore u’(x) = 6 and v(x) = e^{x}

Then we can substitute in one last time in order to get ∫6xe^{x}dx

∫x^{3}e^{x}dx = x^{3}e^{x} – ∫3x^{2}e^{x}dx = x^{3}e^{x} – (3x^{2}e^{x} – (6xe^{x} – ∫6e^{x}dx)) =

x^{3}e^{x} – 3x^{2}e^{x} + 6xe^{x} – 6e^{x} + C. Whew!

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