Integration

1. Integration In recent years Question 1 of the Extension 2 paper has been devoted to integration. These are mostly text book type questions and they should represent good value for the well prepared student. 1. Change of Variable Recently the required substitution has been given in about half of the questions. If the substitution is not given and you suspect that a change of variable is required, look for the presence of a function and its derivative. e.g. sin x and cos x Remember to change the limits of integration

2. Examples:

3. Examples:

6. 2. Integration by Parts Since 1999 most of the questions of this type of integral have had the instruction “use integration by parts” If this instruction is not given but you suspect that you need integration by parts, look for; • The product of two functions where: • (i) one term can be differentiated out, e.g. the log x in xlog x OR (ii) terms that repeat with differentiation, e.g. sin x b) Single functions that you know how to differentiate

7. Examples:

8. Examples:

13. (iii) Trigonometric Integrals (powers of sin x, cos x and tan x) For integrals involving powers of sin x and/or cos x we look for an odd power to become part of the “du” term when substituting u = sin x or u = cos x Examples

14. (iii) Trigonometric Integrals (powers of sin x, cos x and tan x) For integrals involving powers of sin x and/or cos x we look for an odd power to become part of the “du” term when substituting u = sin x or u = cos x Examples

17. If there are only even powers of sin x orcos x, use the fact that Examples

18. If there are only even powers of sin x orcos x, use the fact that Examples

21. For integrals involving tan x or sec x, we use the relationship and the fact that If there is an odd power of sec x, use integration by parts. (similar procedures are used for integrals involving cot x and/or cosec x) Examples

22. For integrals involving tan x or sec x, we use the relationship and the fact that If there is an odd power of sec x, use integration by parts. (similar procedures are used for integrals involving cot x and/or cosec x) Examples

25. (iv) Using the t results This substitution is generally used for integrals of the form This type of integral has only appeared a couple of times since 1999, on each occasion students were given the instruction; “use the substitution ” You need to know that; and that ;

26. Example

27. Example

28. (v) Reduction Formula Here we use integration by parts to generate a formula in terms of an integer, n. This type of question has appeared five times since 1999. Each time the question has been in the second half of the paper and is presented as “Show that …” Example Let n be a positive integer and let (i) Prove that

29. (v) Reduction Formula Here we use integration by parts to generate a formula in terms of an integer, n. This type of question has appeared four times since 1999. Each time the question has been in the second half of the paper and is presented as “Show that …” Example Let n be a positive integer and let (i) Prove that

30. (ii) Hence evaluate

31. (ii) Hence evaluate

32. (vi) Trig Substitutions Example

33. (vi) Trig Substitutions Example

35. (vii) Rational Functions For integrals of the type , if deg P(x) deg Q(x), we must first perform a polynomial division. Then; a) If Q(x) is a linear function, the integral will produce a log function b) If Q(x) is an irreducible quadratic, the integral will produce a log function and/or an integral that requires “completion of the square” c) If Q(x) can be factorised (or is already factorised), use the method of partial fractions. The method of partial fractions appears in most papers, and it usually a two part question with the format of the partial fractions shown in the first part. Integrals that require “completion of the square” have appeared in 8 of the last 10 HSC papers, and, apart from last year came with the instruction; “by completing the square…”

36. Examples

37. Examples

40. 3. (i) Find constants a, b and c such that

41. 3. (i) Find constants a, b and c such that

42. (ii) Hence find

43. (ii) Hence find

44. (viii) Special Properties Examples

45. (viii) Special Properties Examples

48. Volumes In recent HSC papers there has been at least one and generally two questions each year. The method needed is usually stated or implied. In questions involving the technique of slicing there is often a part which states; “show that the cross-sectional area is given by …” This enables you to attempt the final part of the question even if you may not have been able to derive the required expression. Steps In Volume Problems 1. Draw a large diagram, including the slice or shell 2. Find the cross-sectional area of the slice or shell 3. Find the volume of the slice or shell 4. Sum the volumes of the slices (shells) and write as an integral 5. Evaluate the integral

49. 1. Slicing (parallel to the axis of rotation) Try to find an expression for the volume of a slice in terms of the same variable as the thickness of the slice.x or y. Example The region bounded by the curve y = x(4 – x) and the x axis is rotated about the y axis. Find the volume of the solid of revolution by taking slices perpendicular to the y axis.

50. 1. Slicing (perpendicular to the axis of rotation) Try to find an expression for the volume of a slice in terms of the same variable as the thickness of the slice.x or y. Example The region bounded by the curve y = x(4 – x) and the x axis is rotated about the y axis. Find the volume of the solid of revolution by taking slices perpendicular to the y axis.