Basic Rules of Integration
Overview
As with differentiation, there are some basic rules we can apply when integrating functions. If you are familiar with the material in the first few pages of this section, you should by now be comfortable with the idea that integration and differentiation are the inverse of one another. This means that when we integrate a function, we can always differentiate the result to retrieve the original function. Unfortunately, the reverse is not true. Once we differentiate a function, any constant term in that function simply vanishes, because the derivative of any constant term is zero. This is something we need to keep in mind when we think about how we will go about integrating a function, because it means our answer will always contain a constant of unknown value. We call this constant the constant of integration, C .
We have already talked about the power rule for integration elsewhere in this section. This rule is essentially the inverse of the power rule used in differentiation, and gives us the indefinite integral of a variable raised to some power. Just to refresh your memory, the integration power rule formula is as follows:
| ∫ | ax n dx = a | x n+1 | + C |
| n+1 |
This formula gives us the indefinite integral of the variable x raised to the power of n, multiplied by the constant coefficient a (note that n cannot be equal to minus one because this would put a zero in the denominator on the right hand side of the formula). This rule alone is sufficient to enable us to integrate polynomial functions of one variable. We simply integrate each term separately - the plus or minus sign in front of each term does not change. The indefinite integrals of some common expressions are shown below. Note that in these examples, a represents a constant, x represents a variable, and e represents Euler's number (approximately 2.7183). Note also that the first three examples in the table are derived from the application of the power rule.
Indefinite integrals of some common functions
A constant value a:
| ∫ | a dx = ax + C |
A variable x:
| ∫ | x dx = | x 2 | + C |
| 2 |
The square of a variable x 2:
| ∫ | x 2 dx = | x 3 | + C |
| 3 |
The reciprocal of a variable 1/x:
| ∫ | 1 | dx = ln (x) + C |
| x |
The exponential function e x:
| ∫ | e x dx = e x + C |
Other exponential functions a x:
| ∫ | a x dx = | a x | + C |
| ln (a) |
The natural logarithm of a variable ln (x):
| ∫ | ln (x) dx = x ln (x) - x + C |
The sine of a variable sin (x):
| ∫ | sin (x) dx = -cos (x) + C |
The cosine of a variable cos (x):
| ∫ | cos (x) dx = sin (x) + C |
The basic rules of integration, which we will describe below, include the power, constant coefficient (or constant multiplier), sum, and difference rules. We will provide some simple examples to demonstrate how these rules work.
The power rule
The power rule for integration, as we have seen, is the inverse of the power rule used in differentiation. It gives us the indefinite integral of a variable raised to a power. Here is the power rule once more:
| ∫ | ax n dx = a | x n+1 | + C |
| n+1 |
Let's look at a couple of examples of how this rule is used. Suppose we want to find the indefinite integral of x 3. Applying the power rule, we get:
| ∫ | x 3 dx = | x 4 | + C |
| 4 |
Sometimes it is not quite so obvious that we can use the power rule to find the indefinite integral of a function. Suppose, for example, that we want to find the indefinite integral of the expression 3√x. How do we use the power rule to integrate the cubed root function? It's actually quite easy. All we need to do is to rewrite the expression so that we get x to a power. There is a standard formula that allows us to express the nth root of a number a in index form (i.e. as a raised to a power):
n√a = a 1/n
Applying this formula to 3√x we get:
3√x = x 1/3
We can now apply the power rule to get:
| ∫ | x 1/3 dx = | 3x 4/3 | + C |
| 4 |
The constant coefficient rule
The constant coefficient rule (sometimes called the constant multiplier rule) essentially tells us that the indefinite integral of c · ƒ(x), where ƒ(x) is some function and c represents a constant coefficient, is equal to the indefinite integral of ƒ(x) multiplied by c. We can express this formally as follows:
| ∫ | c ƒ(x) dx = c | ∫ | ƒ(x) dx |
The constant coefficient rule essentially allows us to ignore the constant coefficient in an expression while we integrate the rest of the expression. For example, let's suppose we want to find the indefinite integral of the expression 5x 2. The constant coefficient rule tells us that the indefinite integral of this expression is equal to the indefinite integral of x 2 multiplied by five. In other words:
| ∫ | 5x 2 dx = 5 | ∫ | x 2 dx |
Now we just apply the power rule to x 2:
| 5 | ∫ | x 2 dx = | 5x 3 | + C |
| 3 |
The sum rule
The sum rule tells us how we should integrate functions that are the sum of several terms. It basically tells us that we must integrate each term in the sum separately, and then just add the results together. The order in which the terms appear in the result is not important. We can state this formally as follows:
| ∫ | (ƒ(x) + g(x)) dx = | ∫ | ƒ(x) dx + | ∫ | g(x) dx |
You may be wondering at this point why the rule is written in the way that it is. It is quite important to realise here that, in a function that is the sum of two (or more) terms, each term can be considered to be a function in its own right - even a constant term. Suppose we want to find the indefinite integral of the polynomial function ƒ(x) = 6x 2 + 8x + 10. Applying the sum rule, we get:
| ∫ | (6x 2 + 8x + 10) dx = | ∫ | 6x 2 dx + | ∫ | 8x dx + | ∫ | 10 dx |
| ∫ | (6x 2 + 8x + 10) dx = 2x 3 + 4x 2 + 10x + C |
The difference rule
The difference rule tells us how we should integrate functions that involve the difference of two or more terms. It is essentially the same as the sum rule in that it tells us that we must integrate each term in the sum separately. The only difference is that the order in which the terms appear is critical, and must not be changed. We can state this rule formally as follows:
| ∫ | (ƒ(x) - g(x)) dx = | ∫ | ƒ(x) dx - | ∫ | g(x) dx |
Let's look at an example. Suppose we want to find the indefinite integral of the polynomial function ƒ(x) = 4x 3 - 18x - 7. Applying the sum rule, we get:
| ∫ | (4x 3 - 18x - 7) dx = | ∫ | 4x 3 dx - | ∫ | 18x dx - | ∫ | 7 dx |
| ∫ | (6x 2 + 8x + 10) dx = x 4 - 9x 2 - 7x + C |
The sum and difference rules are essentially the same rule. If we want to integrate a function that contains both the sum and difference of a number of terms, the main points to remember are that we must integrate each term separately, and be careful to conserve the order in which the terms appear. The plus or minus sign in front of each term does not change. Alternatively, you can think of the function as the sum of a number of positive and negative terms, and just apply the sum rule. Order is then unimportant - you just need to be mindful of the sign of each term.