1. Chapter 5 Techniques of Integration

2. Chapter 5 Techniques of Integration
5.1 Integration By Parts
If f and g are differentiable functions of x, the Product Rule says
Which implies
Or, equivalently, as
(1)
The application of this formula is called integration by parts.

3. In practice, we usually rewrite (1) by letting
This yields the following alternative form for (1)

4. Example: Use integration by parts to evaluate
Solution:

5. Guidelines for Integration by Parts
The main goal in integration by parts is to choose u and dv to obtain a new integral
That is easier to evaluate than the original.
A strategy that often works is to choose u and dv so that u becomes “simpler” when
Differentiated, while leaving a dv that can be readily integrated to obtain v.
There is another useful strategy for choosing u and dv that can be applied when the
Integrand is a product of two functions from different categories in the list.
Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential
In this case, you will often be successful if you take u to be the function whose
Category occurs earlier in the list and take dv to the rest of the integrand (LIATE).
This method does not work all the time, but it works often enough to b e useful.

6. Example. Evaluate
Solution:

7. Example: Evaluate
Solution:

8. Repeated Integration by Parts
Example: Evaluate
Solution:

10. Example: Evaluate
Solution:

12. Integration by Parts for Definite Integrals
For definite integrals, the formula corresponding to is

13. Example: Evaluate
Solution:

14. 5.2 Trigonometric Integrals
Integrating powers of sine and cosine
By applying the integration by parts , we have two reduction formulas

15. In particular, ……

16. Integrating Products of Sines and Cosines

18. Example: Evaluate
Solution:

19. Examples
Example: Evaluate

20. Integrating Powers of Tangent and Secant
There are similar reduction formulas to integrate powers of tangent and secant.

21. In particular, ……

22. Integrating Products of Tangents and Secents

23. Examples
Example Evaluate

24. Eliminating Square Roots (optional)
Example: Evaluate

25. Section 5.3 Trigonometric Substitutions
We are concerned with integrals that contain expressions of the form
In which a is a positive constant. The idea is to make a substitution for x that will eliminate the radical.
For example, to eliminate the radical in , we can make the substitution
x = a sinx, - /2    /2
Which yields

26. The most common substitutions are x=a tan  , x=a sin  , and x=asec
The most common substitutions are x=a tan  , x=a sin  , and x=asec. They
Come from the reference right triangles below.

27. We want any substitution we use in an integration to be reversible so that we can
Change back to the original variable afterward.

28. Method of Trigonometric Substitution

29. Example
Example 1. Evaluate
Solution.

30. Example Cont.

31. Examples
Example

32. Example
Example. Evaluate , assuming that x5.
Solution. Thus,

33. Example (optional)
There are two methods to evaluate the definite integral.
Make the substitution in the indefinite integral and then evaluate the definite integral using the x-limits of integration.
Make the substitution in the definite integral and convert the x-limits to the corresponding -limits.
Example. Evaluate
Solution:

34. 8.5 Integrating Rational Functions by Partial Fractions
Suppose that P(x) / Q(x) is a proper rational function, by which we mean that the
Degree of the numerator is less than the degree of the denominator. There is a
Theorem in advanced algebra which states that every proper rational function can
Be expressed as a sum
Where F1(x), F2(x), …, Fn(x) are rational functions of the form
In which the denominators are factors of Q(x). The sum is called the partial fraction
Decomposition of P(x)/Q(x), and the terms are called partial fraction.

35. Finding the Form of a Partial Fraction Decomposition
The first step in finding the form of the partial fraction decomposition of a proper
Rational function P(x)/Q(x) is to factor Q(x) completely into linear and irreducible
Quadratic factors, and then collect all repeated factors so that Q(x) is expressed as
A product of distinct factors of the form
From these factors we can determine the form of the partial fraction decomposition
Using two rules that we will now discuss.

36. Linear Factors
If all of the factors of Q(x) are linear, then the partial fraction decomposition of
P(x)/Q(x) can be determined by using the following rule:

37. Example: Evaluate
Solution:

39. Example: Evaluate
Solution:

41. Quadratic Factors
If some of the factors of Q (x) are irreducible quadratics, then the contribution of
Those facctors to the partial fraction decomposition of P(x)/Q(x) can be determined
From the following rule:

42. Examples
Example. Evaluate
Solution.

43. Example Cont.

45. 5.7 Improper Integrals
In this section, we will extend the concept of a definite integral to allow for infinite
Intervals of integration and integrands with vertical asymptotes within the interval of
Integration. We will call the vertical asymptotes infinite discontinuities, and call the
Integrals with infinite intervals of integration or infinite discontinuities within the interval
Of integration improper integrals. Here are some examples:
Improper integrals with infinite limits of integration:
Improper integrals with infinite discontinuities in the interval of integration:
Improper integrals with infinite discontinuities and infinite limits of integration:
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48. Infinite Limits of Integration
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49. Example: Evaluate
Solution:
Example: Evaluate
Solution:
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50. Example: For what values of p does the integral converge?
Solution
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51. Example: Evaluate
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52. Integrands with Vertical Asymptotes
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54. Example: Evaluate
Solution:
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55. Example: Evaluate
Solution:
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56. Example: Evaluate
Solution:
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57. Tests for Convergence and Divergence
When we cannot evaluate an improper integral directly, we try to determine if it converges or diverges. The principal tests for convergence or divergence are the Direct Comparison Test and the Limit Comparison Test.

59. Examples
Example: Decide if converges. Example: Decide if converges