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Integration: Summary
Philippe B. Laval
KSU
September 14, 2005
Abstract This handout summarizes the various integration techniques. It also give alternatives for finding definite integrals when an antiderivative can- not be found.
1 Strategy for Integration
We have studied the following techniques:
- Fundamental theorem of Calculus. This changes the problem of finding an integral to the problem of finding antiderivatives.
- Substitution.
- Integration by parts.
- Partial fraction decomposition.
- Integrals involving trigonometric functions.
- Trigonometric substitution.
- Tables of integrals.
Besides knowing all the above techniques, students must also know the fol- lowing antiderivatives:
undu =
un+ n + 1
u
du = ln |u| + C
eudu = eu^ + C
audu =
au ln a
+ C
sin udu = − cos u + C
cos udu = sin u + C
sec^2 udu = tan u + C
csc^2 udu = − cot u + C
sec u tan udu = sec u + C
csc u cot udu = − csc u + C
sec udu = ln |sec u + tan u| + C
tan udu = ln |sec u| + C
u^2 + 1
du = tan−^1 u + C
1 − u^2
du = sin−^1 u + C
The problem is: given an integral, which method should be used to find it. Unfortunately, there is not a magical recipe. A lot comes from experience. Also, when doing problems, look at which technique worked for a given problem, try to remember it and also try to understand why it worked. The following provides some guidelines regarding which methods to try depending on the type of integral given. Given an integral
f (x) dx ( f (x) is called the integrand), try the following steps:
- Simplify the integrand. Before you try to integrate a function, make sure it is written in its simplest form. This can be achieved by using the various identities studied up to this point.
- Look for an obvious substitution. Try to find some function g (x) in the integrand whose differential g′^ (x) dx also occurs (up to a constant). In this case, try the substitution u = g (x). For example, given
∫ (^) x x^2 − 1
dx, we note that if u = x^2 − 1 , then du = 2xdx. So, we see that both u and du (except for
) then we would try u = x^2 − 1.
- Classify the integrand according to its form. If steps 1 and 2 have failed, try the following:
(a) Trigonometric functions. Try the techniques described in the hand- out on integrals involving trigonometric functions. (b) Rational functions. Try the techniques described in the handout on partial fractions.
2 Approximation of Definite Integrals
By elementary functions, we mean polynomial, rational, power, exponential, logarithmic, trigonometric, and inverse trigonometric functions and functions which can be obtained from these by addition, subtraction, multiplication, di- vision and composition. These are the functions students are more likely to encounter in traditional calculus classes. The question then becomes can we integrate all the elementary functions? The answer is no. A simple function such as ex 2 does not have an antiderivative which can be expressed in terms of elementary functions. Thus, we cannot evaluate
ex 2 dxin terms of functions we know. The same is true for the following:
∫ (^) ex x
dx
sin
x^2
dx
cos (ex) dx
∫ (^) dx ln x
∫ (^) sin x x dx
What do we do when a problem arises in which we have to evaluate an integral containing one of these functions? We know the answer. We know we can approximate integrals using Riemann sums. In other words,
∫ (^) b
a
f (x) dx =
b − a n
∑^ n
i=
f (x∗ i )
In the above formula, the following is assumed:
- The interval [a, b] has been subdivided into n sub-intervals of equal length.
- x∗ i is a point selected in the ith sub-interval.
Though x∗ i can be selected anywhere, we usually consider the following five cases:
- x∗ i is the left endpoint of the interval in which it is selected. We call this sum the left Riemann sum and denote it Ln.
- x∗ i is the right endpoint of the interval in which it is selected. We call this sum the right Riemann sum and denote it Rn.
- x∗ i is the midpoint of the interval in which it is selected. We denote this sum Mn.
- x∗ i corresponds to the maximum value of f in the interval in which it is selected. This will always give a value larger than the actual value of the integral. We call this sum the upper Riemann sum, and denote it Un.
- x∗ i corresponds to the minimum value of f in the interval in which it is selected. This will always give a value smaller than the actual value of the integral. We call this sum the lower Riemann sum and denote it Sn.
For further details on how this is done, consult the handout on the definite integral given at the beginning of the semester. We will always have
Sn ≤
∫ (^) b
a
f (x) dx ≤ Un
This gives us an upper bound and a lower bound on the value of the integral. By taking n larger and larger, we will get a better and better approximation. The Riemann sum applet, which can be found at
http://science.kennesaw.edu/~plaval/applets/Riemann.html
can find these Riemann sums. It should be noted that Riemann sums can be improved to give a better approximation. These techniques will not be discussed here. Some can be found in many calculus books. Others can be found in books on Numerical Analysis.
3 Additional Tools to Find Integrals
Advanced calculators such as the TI 82, TI 83, TI 86, TI 89, TI 92 as well as many computer software (CAS for Computer Algebra Systems) can evaluate integrals. They usually take one of two approaches. These are describes below.
3.1 Symbolic Integration
This refers to integration by finding an antiderivative first, then using the limits of integration if any are provided. Systems which can perform symbolic integra- tion can find indefinite integrals such as
f (x) dx as well as definite integrals
such as
∫ (^) b a f^ (x)^ dx. They will give an exact answer. Unfortunately, this is very difficult to do, and few machines or programs have this capability. The TI 92 can perform symbolic integration. The following CAS, widely used in the scientific community can also perform symbolic integration:
