10.1 Concept
We define integration as the reverse process of differentiation. That is, the integral with respect to x of a function f (x), denoted
Z
f (x)dx is F (x) if
d
dx(F (x)) = f (x).
Note that the dx in the integral is not something to be integrated, rather it is the end of the notation and denotes something like “with respect to x”
in this context.
10.1.1 Constant of Integration
Consider that d
dx(5x − 10) = d
dx(5x) = d
dx(5x + 1000) = 5
If we consider the problem of finding the integral of 5 with respect to x we need something that differentiates to be 5, so any of these functions could do.
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When we differentiate, any constant present is lost, and therefore when we integrate we assume that a constant may have been present. This is called the constant of integration, and is usually denoted by c. Thus
Z
5dx = 5x + c.
10.2 Rules & Techniques
Differentiation is a relatively simple procedure; with practice we can differen-tiate almost any function. Integration is, in general, much harder and many of our favourite rules have no equivalent, so we must use a variety of less perfect techniques. As before we look at some rules and techniques before introducing examples.
10.2.1 Power Rule
Integrating powers of x is almost as simple as differentiating:
Z
xndx = xn+1
n + 1+ c; (n 6= −1)
So we raise the power by one and divide by the new power. Note that this rule breaks down in one case, when n = −1, which corresponds to the integral of 1x, at this value we wind up with division by zero in our above formula.
We need something that differentiates to this function, and fortunately we met something like this in differentiation.
Z
x−1dx = Z 1
xdx = ln x + c So that all powers of x are covered.
10.2.2 Addition & Subtraction
Suppose that u and v are functions of x. Then Z
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In other words, we can split integration up over addition and subtraction, just as we could do with differentiation.
10.2.3 Multiplication by a constant
Suppose that u is a function of x and k is a constant. Then Z
kudx = k Z
udx.
In other words, we can take constants outside integrals, just as we could do with differentiation.
10.2.4 Substitution
We have no analog of the chain rule in integration, but the closest thing is the idea of substitution.
Once again, we look for an ackward expression that would be easier if it were x, and we let this be u. We then differentiate u with respect to x, and imagine that we can split the du from the dx. We rearrange for dx (and sometimes other bits of x terms and replace the dx with du. We then perform the u based integral.
10.2.5 Limited Chain Rule
When we looked at the chain rule in differentiation (see 9.3.4 we saw that there was a quick way to think about it. We can differentiate the object u as if it were x, provided we multiply by dudx. For example, we have the basic
rule d
dx(sin x) = cos x, but the chain rule can extend this to
d
dx(sin u) = cos u × du dx.
This is very quick to operate, and the laborious process of subsitution in integration does not easily lend itself to this speed.
Fortunately, if our u = ax + b where a and b are constants (b may be zero), we can take a shortcut. We integrate u just as we would x, and then
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divide by a to compensate. So for example, we have the basic integration
rule Z
1
xdx = ln x + c
We cannot extend this to anything like we were able to with the chain rule in differentiation, but if the thing replacing x is simple, namely ax + b we can. For example:
Z 1
ax + bdx = 1
aln(ax + b) + c,
and we can extend every integration result in this way, provided we only replace x by ax + b.
We shall see later that Z
exdx = ex+ c and so
Z
eax+bdx = 1
aeax+b+ c and so on.
10.2.6 Logarithm rule
Again, appealing to the differentiation chain rule (see 9.3.4) we can extend the basic result
d
dx(ln x) = 1 x
to d
dx(ln u) = 1 u × du
dx = u0 u where u is some function of x. It follows that
Z f0(x)
f (x)dx = ln f (x) + c.
That is, if we need to integrate a function where the numerator is the deriva-tive of the denominator we simply state it to be the natural logarithm of the bottom line (plus the usual constant of course).
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10.2.7 Partial Fractions
Because of the lack of solid rules in integration, we must rely with our inge-nuity in algebra far more than was the case in differentiation. One technique that is useful for dealing with fraction expressions is called partial fractions in which we try to split a complex fraction into several smaller, simpler ones.
The first step of the method is to attempt to factorize the bottom line as completely as possible, and the exact action will be decided partially on how well the bottom factors, and on the degree of the top line of the equation.
We shall assume that the degree1 of the top line is always less than the degree of the lower line. If this is not the case we must begin with long division in algebra which is beyond the scope of this module.
We consider several cases and in each case we give a suitable right hand side which we hope to simplify our expression to.
Denominator factors completely into linear factors
In this case, the bottom line of the expression can be decomposed entirely into linear factors, that is, factors of the form (ax + b) like (2x − 3), or 3x etc.. In this case we shall use the following target expression.
gx + h
(ax + b)(cx + d) = A
ax + b + B cx + d
So the expression on the right hand side is formed entirely of numbers over each factor. The numbers are represented by the capitals A and B and are to be determined.
Denominator factors with repeated linear factors
This is similar to the previous case in that the bottom line decomposes com-pletely into linear factors, but this time one of those factors is repeated. For example
gx + h (ax + b)(cx + d)2
In this case it is not correct to use the expected expansion gx + h
(ax + b)(cx + d)2 = A
ax + b + B
cx + d + C cx + d
1The degree of an equation is the index of the highest power of x, thus a quadratic equation is of degree 2.
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as the bottom lines on the right will not give the correct denominator when a common denominator is taken. Instead we list a constant over the factor, and another constant over the factor squared, and so on until the power of the factor is that we began with. In this case the factor was squared so the correct treatment would be: Note the square on the bottom of the last bracket.
Denominator factors incompletely
Sometimes one factor on the bottom line will not factorize completely. For example if we get a factor of (x2 + 1) then this cannot be factorized into linear form (as the quadratic has no solutions). We shall only consider the case that one factor is a quadratic that cannot be factorized further.
gx + h
(ax + b)(cx2+ dx + e) = A
ax + b + Bx + C cx2 + dx + e Method
Once the correct form for simplification has been established as discussed above we need to determine the constants A, B, . . . .
Note that the left hand side and the right hand side are identically equal.
That is to say, they match for all values of x. One way of establishing the values of A, B, . . . , is to insert values of x repeatedly. Each of these will produce an equation. So if we have constants A, B and C only then we will need three equations. In practice however, we can pick strategic values of x to make this easier.
10.2.8 Integration by Parts
The integration by parts formula is the closest thing that we have to a product rule for differentiation, and it is nothing like as powerful.
If u and v are functions of x then
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Proof
We include a simple proof for interest and completeness, it is not examinable.
This result comes from the differentiation product rule:
d
dx(uv) = vdu
dx + udv dx Integrating with respect to x both sides obtains
uv = Z
vdu dxdx +
Z udv
dxdx,
with the derivative and integral cancelling on the LHS. We now rearrange this to the parts formula by, for example, subtracting R udvdxdx on both sides.
Observations
Note that to use this as a product rule we need to pick one bit of the product to be v and the other to be dudx. We will also need to find dvdx and u. It follows that we must pick one bit we can differentiate (v) and one bit we can integrate dudx.
The RHS of the formula consists of two terms; the term uv is of no concern, it is already integrated, but the other term can be a problem. If we cannot work out the integral on the RHS this method is no use. We must make this integral easier than our first one, or at least no worse.
10.2.9 Other rules
There are some other rules in integration that are useful, which are as follows.
Trigonometric functions Z
sin xdx = − cos x + c Z
cos xdx = sin x + c Z
tan xdx = ln | sec x| + c Z
cot xdx = ln | sin x| + c
10.3 Examples 179
Here are some examples of various integrations.