1. Techniques of Integration
Chapter 7
Techniques of Integration

2. 7.1 Integration by parts Question: How to integrate
, where the integrands are the product of two kinds of functions?
Every differentiation rule has a corresponding integration rule:
Differentiation Integration
the Chain Rule the Substitution Rule
the Product Rule the Rule for Integration by Parts

3. The formula for integration by parts (1)
Let u = f (x) and v = g(x) are both differentiable, then du= f’(x) dx and dv= g’(x) dx
(2)

4. Example 1. Find
Example 2. Evaluate
Example 3. Find
Example 4. Evaluate

5. The formula for definite integration by parts (3)
Example 5. Calculate
Example 6. Prove the reduction formula
where is an integer.

6. Summarize
When the integrands are the product of two kinds of functions and neither of them is derivative of the other, we use the integration by parts. We can compare:

7. First we recognize u and v, then confirm to be more easily integrated than . For example

8. 7.2 Trigonometric integrals
Question 1: How to evaluate ？
(a) If the power of cosine is odd ( n=2k+1 ).
Save one cosine factor to
Use to express the remaining factors in terms of sine:
Then substitute u=sinx.

9. (b) If the power of sine is odd ( m=2k+1 ).
Save one sine factor to
Use to express the remaining factors in terms of sine:
Then substitute u=cosx.
(c) If the powers of both sine and cosine are even, use the half-angle identities

10. It is sometimes helpful to use the identity
Example 1. Evaluate
Example 2. Evaluate
Example 3. Find

11. Question 2: How to evaluate ？
(a) If the power of secant is even ( n=2k).
Save a factor of
Use to express the remaining factors in terms of tanx:
Then substitute u=tanx.
Example 4. Find

12. (b) If the power of tangent is odd ( m=2k+1).
Save a factor of
Use to express the remaining factors in terms of secx:
Then substitute u=secx.
Example 5. Find

13. (c) If n = 0, only tanx occurs. Use and, if necessary, the formula
Example 6. Find
(d) If n is odd and m is even, we express the integrand completely in term of secx.
Power of secx may require integration by parts.
Example 7. Find
Example 8. Find

14. Question 3: How to evaluate ？
To evaluate the integrals (a)
(b)
use the corresponding identity:
Example 9. Evaluate

15. 7.3 Trigonometric substitution
How to find the area of a circle or an ellipse?
How to integrate
In general we can make a substitution of the form x=g(t) by using the Substitution Rule in reverse(called inverse substitution):
Assume that g has an inverse function, that is, g is one-to-one, we obtain

16. Table of trigonometric substitution
One kind of inverse substitution is trigonometric substitution.
Table of trigonometric substitution
Expression Substitution Identity

17. Example 1. Evaluate
Example 2. Find
Example 3. Evaluate
Example 4. Evaluate

18. Inverse Substitution Formula For Definite Integral
Let x=g( t ) and g has an inverse function, we have
where
Example 5. Find
Example 6. Find the area enclosed by the ellipse
Example 7. Find

19. 7.4 Integration of rational functions
by partial fraction
In this section we show how to integrate any rational function (a ratio of polynomials) by expressing it as a sum of simpler fraction(called partial fraction).
Consider a rational function
where P and Q are polynomials. If
where then the degree of P is n and we write
deg(P) = n

20. Step 1. First express f as a sum of simpler fractions.
We can integrate rational functions according to 3 steps:
Step 1. First express f as a sum of simpler fractions.
Provide that the degree of P is less than the degree of Q. Such a rational function is called proper.
If f is improper, that is, we can divide Q into P by long division until a remainder R(x) is obtained such that deg(R)<deg(Q). The division statement is
(1)
where S and R are also polynomials.

21. Step 2. Second factor the denominator Q(x) as far as possible
Step 2. Second factor the denominator Q(x) as far as possible. It can be shown that any polynomial Q can be factored as a product of linear factors(of the form ax+b) and irreducible quadratic factors (of the form
Step 3. Finally express the proper rational function R(x)/Q(x) as a sum of partial fractions of the form
A theorem in algebra guarantees that it is always possible to do it.

22. Case 1. The denominator Q(x) is a product of distinct linear factors.
We explain the details for the 4 cases that occur.
Case 1. The denominator Q(x) is a product of distinct linear factors.
The partial fraction theorem states there exist constants
such that
(2)
These constants need to be determined.
Example 1. Evaluate

23. Example 2. Find
Case 2. Q(x) is a product of linear factors, some of which are repeated.
Suppose the first linear factor is repeated r times; that is, occurs in the factorization of Q(x). Then instead of the single term
in Equation 2, we would use
(3)
Example 3. Find

24. Case 3. Q(x) contains irreducible quadratic factors, none of which is repeated.
If Q(x) has the factor then, in addition to the partial fraction in Equation 2 and 3, the expression for R(x)/Q(x) will have a term of the form
(4)
where A and B are constants to be determined. We can integrate (4) by completing the square and using the formula
(5)
Example 3. Find

25. Case 4. Q(x) contains a repeated irreducible quadratic factors.
Example 4. Evaluate
Case 4. Q(x) contains a repeated irreducible quadratic factors.
If Q(x) has the factor then instead of the single partial fraction(4), the sum
(6)
occurs in the partial fraction decomposition of R(x)/Q(x). Each of the term in (6) can be integrated by completed the square and making a tangent substitution.
Example 5. Evaluate

26. 7.5 Rationalizing substitutions
By means of appropriate substitutions, some functions can be changed into rational functions. In particular, when an integrand contains an expression of the form , then the substitution u = may be effective.
Example 1. Evaluate
Let
Example 2. Find
Let

27. The substitution t = tan(x/2) will convert any rational function of sinx and cosx into an ordinary rational function. This is called Weierstrass substitution.
Let
Then
Therefore

28. (1)
Since t = tan(x/2), we have , so
Thus if we make the substitution t = tan(x/2), then we have
Example 3. Find

29. 7.6 Strategy For Integration
Integration is more challenging than differentiation.
In finding the derivative of a function it is obvious which differentiation formula we should apply. But when integrating a given function, it may not be obvious which techniques we should use.
First it is useful to be familiar with the basic integration formulas.
Table of integration formulas
Constants of integration have been omitted.

31. 1. Simplify the integrand if possible.
Secondly if you do not immediately see how to attack a given integral, you might try the following four-step strategy.
1. Simplify the integrand if possible.
2. Look for an obvious substitution.
3. Classify the integrand according to its form.
4. Try again.
(a) Try substitution (b) Try parts.
(c) Manipulate the integrand.
(d) Relate the problem to previous problems.
(e) Use several methods.

32. Example 1.
Example 2.
Example 3.
Example 4.
Example 5.

33. Computer Algebra Systems
7.7 Using Tables of Integrals and
Computer Algebra Systems

34. 7.8 Approximation Integration
How to integrate
It is difficult, or even impossible, to find an antiderivative.
When the function is determined from a scientific experiment through instrument readings, how to integrate such discrete function?
In both cases we need to find approximate values of definite integrals.

35. The left endpoint approximation
Using Riemann sums
The left endpoint approximation
(1)
The right endpoint approximation
(2)
(3) Midpoint rule

36. Example 1. Use (a) the Trapezoidal Rule (b) the Midpoint Rule with n=5
to approximate the integral

37. (5) Error bounds Suppose If
Notice
The error in using an approximation is defined to be the amount that needs to be added to the approximation to make it exact. .
In general, we have
(5) Error bounds Suppose If
and are the errors in the Trapezoidal and Midpoint Rules, then

38. Example 2. (a) Use the Midpoint Rule with n=10 to approximate the integral
(b) Give an upper bound for the error involved in this approximation.
(6) Simpson’s Rule
where n is even and
Example 3. Use Simpson’s Rule with n=10 to approximation

39. (7) Error bound for Simpson’s Rule Suppose that
If is the error involved in using Simpson’s Rule, then
Example 4. (a) Use Simpson’s Rule with n=10 to approximate the integral
(b) Estimate the error involved in this approximation.

40. Type 1 Infinite Intervals
7.9 Improper Integrals
In defining a definite integral we deal with (1) the function f defined on a finite interval [a, b]; (2) f is a bounded function.
Question: How to integrate a definite integral when the interval is infinite or when f is unbounded ?
Type 1 Infinite Intervals
Consider the infinite region that lies under the curve , above the x-axis and from the line x=1 to infinite, can this area A be infinite?

41. Notice
and
So the area of the infinite region is equal to 1 and we write

42. (1)Definition of An Improper Integral of Type 1
(a) If exists for every number , then
provide this limit exists(as a finite number).
(b) If exists for every number , then
The improper integrals in (a) and (b) are called convergent if the limit exists and divergent if the limit does not exist.
(c) If both and are convergent, then we define

43. Example 1. Determine whether the integral
is convergent or divergent.
Example 2. Evaluate
Example 3. Evaluate
Example 4. For what value of p is the integral
convergent?
(2) is convergent if p >1 and divergent if

44. Type 2 Discontinuous Integrands
(3) Definition of An Improper Integral of Type 2
(a) If f is continuous on [a,b) and , then
if this limit exists(as a finite number).
(b) If f is continuous on (a,b] and , then
The improper integrals in (a) and (b) are called convergent if the limit exists and divergent if the limit does not exist.

45. (c) If , where a<c<b, and both
and are convergent, then we define
Example 5. Find
Example 6. Determine whether converges or
diverges.
Example 7. Evaluate if possible.
Example 8. Find

46. A Comparison Test For Improper Integrals
(4)Comparison Theorem Suppose that f and g are continuous functions with
(a) If is convergent, then is convergent.
(b) If is divergent, then is divergent.
Example 9. Show that (a) is convergent.
(b) is divergent.