The document discusses trigonometric functions, arcs, sectors, and related concepts. It defines:
- 360° = 2π radians
- The circumference of a circle is given by C = 2πr
- The area of a circle is given by A = πr^2
- The length of an arc is given by l = rθ
- The area of a sector is given by A = (1/2)r^2θ
It provides an example calculating the length of an arc and area of a sector for a circle with radius 5cm and central angle of 45°.
The AllegroGraph WebView (AGWebView) is a user interface for exploring, querying, and managing AllegroGraph triple stores. Using AGWebView, users can browse repositories and data, issue SPARQL and Prolog queries, view and navigate query results, and manage users and access permissions. Key features include the ability to load and reason over RDF data, configure triple indices, define namespaces, and visualize query results on a map or as a graph.
The document discusses trigonometric functions, arcs, sectors, and related concepts. It defines:
- 360° = 2π radians
- The circumference of a circle is given by C = 2πr
- The area of a circle is given by A = πr^2
- The length of an arc is given by l = rθ
- The area of a sector is given by A = (1/2)r^2θ
It provides an example calculating the length of an arc and area of a sector for a circle with radius 5cm and central angle of 45°.
Slideshare is discontinuing its Slidecast feature as of February 28, 2014. Existing Slidecasts will be converted to static presentations without audio by April 30, 2014. The document informs users that new slidecasts can be found on myPlick.com or the author's blog starting in 2014. However, myPlick proved unreliable, so future slidecasts will instead be hosted on the author's YouTube channel.
The document discusses different methods for factorising expressions:
1) Looking for a common factor and dividing it out of all terms
2) Using the difference of two squares formula (a2 - b2 = (a - b)(a + b))
3) Factorising quadratic trinomials into two binomial factors by identifying the values that multiply to give the constant term and sum to give the coefficient of the linear term.
The document provides information on index laws and the meaning of indices in algebra:
- Index laws state that am × an = am+n, am ÷ an = am-n, and (am)n = amn. Exponents can be added or subtracted when multiplying or dividing terms with the same base.
- Positive exponents indicate a term is raised to a power. Negative exponents indicate a root is being taken. Terms with exponents are evaluated from left to right.
- Examples demonstrate how to simplify expressions using index laws and interpret different types of indices.
12 x1 t01 03 integrating derivative on function (2013)Nigel Simmons
The document discusses integrating derivatives of functions. It states that the integral of the derivative of a function f(x) is equal to the natural log of f(x) plus a constant. It then provides examples of integrating several derivatives: (i) ∫(1/(7-3x)) dx = -1/3 log(7-3x) + c, (ii) ∫(1/(8x+5)) dx = 1/8 log(8x+5) + c, and (iii) ∫(x5/(x-2)) dx = 1/6 log(x6-2) + c. It also discusses techniques for integrating fractions by polynomial long division and finds
The document discusses logarithms and their properties. Logarithms are defined as the inverse of exponentials. If y = ax, then x = loga y. The natural logarithm is log base e, written as ln. Properties of logarithms include: loga m + loga n = loga mn; loga m - loga n = loga(m/n); loga mn = n loga m; loga 1 = 0; loga a = 1. Examples of evaluating logarithmic expressions are provided.
The document discusses relationships between the coefficients and roots of polynomials. It states that for a polynomial P(x) = axn + bxn-1 + cxn-2 + ..., the sum of the roots equals -b/a, the sum of the roots taken two at a time equals c/a, and so on for higher order terms. It also provides examples of using these relationships to find the sums of roots for a given polynomial.
P
4
3
2
The document discusses properties of polynomials with multiple roots. It first proves that if a polynomial P(x) has a root x = a of multiplicity m, then the derivative of P(x), P'(x), will have a root x = a of multiplicity m-1. It then provides an example of solving a cubic equation given it has a double root. Finally, it examines a quartic polynomial and shows that its root α cannot be 0, 1, or -1, and that 1/
The document discusses factorizing complex expressions. The main points are:
- If a polynomial's coefficients are real, its roots will appear in complex conjugate pairs.
- Any polynomial of degree n can be factorized into a mixture of quadratic and linear factors over real numbers, or into n linear factors over complex numbers.
- Odd degree polynomials must have at least one real root.
- Examples of factorizing polynomials over both real and complex numbers are provided.
The document describes the Trapezoidal Rule for approximating the area under a curve between two points. It shows that the area A is estimated by dividing the region into trapezoids with height equal to the function values at the interval endpoints and bases equal to the intervals. In general, the area is approximated as the sum of the areas of each trapezoid, which is equal to the average of the endpoint function values multiplied by the interval length.
The document discusses trigonometric functions, arcs, sectors, and related concepts. It defines:
- 360° = 2π radians
- The circumference of a circle is given by C = 2πr
- The area of a circle is given by A = πr^2
- The length of an arc is given by l = rθ
- The area of a sector is given by A = (1/2)r^2θ
It provides an example calculating the length of an arc and area of a sector for a circle with radius 5cm and central angle of 45°.
The AllegroGraph WebView (AGWebView) is a user interface for exploring, querying, and managing AllegroGraph triple stores. Using AGWebView, users can browse repositories and data, issue SPARQL and Prolog queries, view and navigate query results, and manage users and access permissions. Key features include the ability to load and reason over RDF data, configure triple indices, define namespaces, and visualize query results on a map or as a graph.
The document discusses trigonometric functions, arcs, sectors, and related concepts. It defines:
- 360° = 2π radians
- The circumference of a circle is given by C = 2πr
- The area of a circle is given by A = πr^2
- The length of an arc is given by l = rθ
- The area of a sector is given by A = (1/2)r^2θ
It provides an example calculating the length of an arc and area of a sector for a circle with radius 5cm and central angle of 45°.
Slideshare is discontinuing its Slidecast feature as of February 28, 2014. Existing Slidecasts will be converted to static presentations without audio by April 30, 2014. The document informs users that new slidecasts can be found on myPlick.com or the author's blog starting in 2014. However, myPlick proved unreliable, so future slidecasts will instead be hosted on the author's YouTube channel.
The document discusses different methods for factorising expressions:
1) Looking for a common factor and dividing it out of all terms
2) Using the difference of two squares formula (a2 - b2 = (a - b)(a + b))
3) Factorising quadratic trinomials into two binomial factors by identifying the values that multiply to give the constant term and sum to give the coefficient of the linear term.
The document provides information on index laws and the meaning of indices in algebra:
- Index laws state that am × an = am+n, am ÷ an = am-n, and (am)n = amn. Exponents can be added or subtracted when multiplying or dividing terms with the same base.
- Positive exponents indicate a term is raised to a power. Negative exponents indicate a root is being taken. Terms with exponents are evaluated from left to right.
- Examples demonstrate how to simplify expressions using index laws and interpret different types of indices.
12 x1 t01 03 integrating derivative on function (2013)Nigel Simmons
The document discusses integrating derivatives of functions. It states that the integral of the derivative of a function f(x) is equal to the natural log of f(x) plus a constant. It then provides examples of integrating several derivatives: (i) ∫(1/(7-3x)) dx = -1/3 log(7-3x) + c, (ii) ∫(1/(8x+5)) dx = 1/8 log(8x+5) + c, and (iii) ∫(x5/(x-2)) dx = 1/6 log(x6-2) + c. It also discusses techniques for integrating fractions by polynomial long division and finds
The document discusses logarithms and their properties. Logarithms are defined as the inverse of exponentials. If y = ax, then x = loga y. The natural logarithm is log base e, written as ln. Properties of logarithms include: loga m + loga n = loga mn; loga m - loga n = loga(m/n); loga mn = n loga m; loga 1 = 0; loga a = 1. Examples of evaluating logarithmic expressions are provided.
The document discusses relationships between the coefficients and roots of polynomials. It states that for a polynomial P(x) = axn + bxn-1 + cxn-2 + ..., the sum of the roots equals -b/a, the sum of the roots taken two at a time equals c/a, and so on for higher order terms. It also provides examples of using these relationships to find the sums of roots for a given polynomial.
P
4
3
2
The document discusses properties of polynomials with multiple roots. It first proves that if a polynomial P(x) has a root x = a of multiplicity m, then the derivative of P(x), P'(x), will have a root x = a of multiplicity m-1. It then provides an example of solving a cubic equation given it has a double root. Finally, it examines a quartic polynomial and shows that its root α cannot be 0, 1, or -1, and that 1/
The document discusses factorizing complex expressions. The main points are:
- If a polynomial's coefficients are real, its roots will appear in complex conjugate pairs.
- Any polynomial of degree n can be factorized into a mixture of quadratic and linear factors over real numbers, or into n linear factors over complex numbers.
- Odd degree polynomials must have at least one real root.
- Examples of factorizing polynomials over both real and complex numbers are provided.
The document describes the Trapezoidal Rule for approximating the area under a curve between two points. It shows that the area A is estimated by dividing the region into trapezoids with height equal to the function values at the interval endpoints and bases equal to the intervals. In general, the area is approximated as the sum of the areas of each trapezoid, which is equal to the average of the endpoint function values multiplied by the interval length.
The document discusses methods for calculating the volumes of solids of revolution. It provides formulas for finding volumes when an area is revolved around either the x-axis or y-axis. Examples are given for finding volumes of common solids like cones, spheres, and others. Steps are shown for using the formulas to calculate volumes based on given functions and limits of revolution.
The document discusses different methods for calculating the area under a curve or between curves.
(1) The area below the x-axis is given by the integral of the function between the bounds, which can be positive or negative depending on whether the area is above or below the x-axis.
(2) To calculate the area on the y-axis, the function is solved for x in terms of y, then the bounds are substituted into the integral of this new function with respect to y.
(3) The area between two curves is calculated by taking the integral of the upper curve minus the integral of the lower curve, both between the same bounds on the x-axis.
The document discusses 8 properties of definite integrals:
1) Integrating polynomials results in a fraction.
2) Constants can be factored out of integrals.
3) Integrals of sums are equal to the sum of integrals.
4) Splitting an integral range results in the sum of the integrals.
5) Integrals of positive functions over a range are positive, and negative if the function is negative.
6) Integrals can be compared based on the relative values of the integrands.
7) Changing the limits of integration flips the sign of the integral.
8) Integrals of odd functions over a symmetric range are zero, and integrals of even functions are twice the integral over
0
The document discusses the relationship between integration and calculating the area under a curve. It shows that the area under a curve from x=a to x=b can be calculated as the integral from a to b of the function f(x) dx. This is equal to the antiderivative F(x) evaluated from b to a. The area under a curve can also be estimated using rectangles, and as the width of the rectangles approaches 0, the estimate becomes the exact area. The derivative of the area function A(x) gives the equation of the curve f(x). Examples are given to calculate the exact area under curves.
4. Trig Integrals
(1) Standard Integrals
1
sin axdx a cos ax c
1
cos axdx a sin ax c
1
sec 2 axdx tan ax c
a
5. Trig Integrals
(1) Standard Integrals
1
sin axdx a cos ax c
1
cos axdx a sin ax c
1
sec 2 axdx tan ax c
a
sin ax
tan axdx cos ax dx
6. Trig Integrals
(1) Standard Integrals
1
sin axdx a cos ax c
1
cos axdx a sin ax c
1
sec 2 axdx tan ax c
a
sin ax
tan axdx cos ax dx
1 1
log cos ax c OR log sec ax c
a a
9. 2 sin n x or cos n x
sin xdx cos x c
sin 2 xdx
10. 2 sin n x or cos n x
sin xdx cos x c
1
sin xdx 2 1 cos 2 x dx
2
11. 2 sin n x or cos n x
sin xdx cos x c
1
sin xdx 2 1 cos 2 x dx
2
1 1
x sin 2 x c
2 2
12. 2 sin n x or cos n x
sin xdx cos x c
1
sin xdx 2 1 cos 2 x dx
2
1 1
x sin 2 x c
2 2
sin 3 xdx
13. 2 sin n x or cos n x
sin xdx cos x c
1
sin xdx 2 1 cos 2 x dx
2
1 1
x sin 2 x c
2 2
sin 3 xdx sin x sin xdx
2
14. 2 sin n x or cos n x
sin xdx cos x c
1
sin xdx 2 1 cos 2 x dx
2
1 1
x sin 2 x c
2 2
sin 3 xdx sin x sin xdx
2
sin x1 cos 2 x dx
15. 2 sin n x or cos n x
sin xdx cos x c
1
sin xdx 2 1 cos 2 x dx
2
1 1
x sin 2 x c
2 2
sin 3 xdx sin x sin xdx
2
sin x1 cos 2 x dx u cos x
du sin xdx
16. 2 sin n x or cos n x
sin xdx cos x c
1
sin xdx 2 1 cos 2 x dx
2
1 1
x sin 2 x c
2 2
sin 3 xdx sin x sin xdx
2
sin x1 cos 2 x dx u cos x
du sin xdx
1 u 2 du
17. 2 sin n x or cos n x
sin xdx cos x c
1
sin xdx 2 1 cos 2 x dx
2
1 1
x sin 2 x c
2 2
sin 3 xdx sin x sin xdx
2
sin x1 cos 2 x dx u cos x
du sin xdx
1 u 2 du
1 3
u u c
3
1 3
cos x cos x c
3
18. 2 sin n x or cos n x
sin xdx cos x c
1
sin xdx 2 1 cos 2 x dx
2
1 1
x sin 2 x c
2 2
sin 3 xdx sin x sin xdx
2
sin x1 cos 2 x dx u cos x
du sin xdx
1 u 2 du
Odd Power
1 3
u u c Factorise as sin xsin x
2 some power
3
1 3 Substitute sin 2 x 1 cos 2 x
cos x cos x c
3 Use u cos x
21. sin xdx sin x dx
4 2 2
1
1 cos 2 x dx
2
4
22. sin xdx sin x dx
4 2 2
1
1 cos 2 x dx
2
4
1 2 cos 2 x cos 2 2 x dx
1
4
23. sin xdx sin x dx
4 2 2
1
1 cos 2 x dx
2
4
1 2 cos 2 x cos 2 2 x dx
1
4
1 2 cos 2 x 1 cos 4 x dx
1 1
4 2
24. sin xdx sin x dx
4 2 2
1
1 cos 2 x dx
2
4
1 2 cos 2 x cos 2 2 x dx
1
4
1 2 cos 2 x 1 cos 4 x dx
1 1
4 2
2 cos 2 x cos 4 x dx
1 3 1
4 2 2
25. sin xdx sin x dx
4 2 2
1
1 cos 2 x dx
2
4
1 2 cos 2 x cos 2 2 x dx
1
4
1 2 cos 2 x 1 cos 4 x dx
1 1
4 2
2 cos 2 x cos 4 x dx
1 3 1
4 2 2
x sin 2 x sin 4 x c
13 1
42 8
26. xdx sin x dx
Even Power
sin
4 2 2
Factorise as sin x
2 some power
1
1 cos 2 x dx
2
1
4 Substitute sin x 1 cos 2 x
2
2
1 2 cos 2 x cos 2 2 x dx
1
4
1 2 cos 2 x 1 cos 4 x dx
1 1
4 2
2 cos 2 x cos 4 x dx
1 3 1
4 2 2
x sin 2 x sin 4 x c
13 1
42 8
29. sin xdx sin xsin x dx
5 2 2
sin x1 cos x dx
2 2
30. sin xdx sin xsin x dx
5 2 2
sin x1 cos x dx u cos x
2 2
du sin xdx
31. sin xdx sin xsin x dx
5 2 2
sin x1 cos x dx u cos x
2 2
du sin xdx
1 u du
2 2
1 2u 2 u 4 du
32. sin xdx sin xsin x dx
5 2 2
sin x1 cos x dx u cos x
2 2
du sin xdx
1 u du
2 2
1 2u 2 u 4 du
u 2 u3 1 u5 c
3 5
2 1
cos x cos3 x cos 5 x c
3 5
34. 3 sin n x and cos n x
Usually done by substitution u sin x or u cos x
35. 3 sin n x and cos n x
Usually done by substitution u sin x or u cos x
e.g. i cos 5 x sin 3 xdx
36. 3 sin n x and cos n x
Usually done by substitution u sin x or u cos x
e.g. i cos 5 x sin 3 xdx
cos5 x1 cos 2 x sin xdx
37. 3 sin n x and cos n x
Usually done by substitution u sin x or u cos x
e.g. i cos 5 x sin 3 xdx
cos5 x1 cos 2 x sin xdx u cos x
du sin xdx
38. 3 sin n x and cos n x
Usually done by substitution u sin x or u cos x
e.g. i cos 5 x sin 3 xdx
cos5 x1 cos 2 x sin xdx u cos x
u 5 1 u 2 du
du sin xdx
u 7 u 5 du
39. 3 sin n x and cos n x
Usually done by substitution u sin x or u cos x
e.g. i cos 5 x sin 3 xdx
cos5 x1 cos 2 x sin xdx u cos x
u 5 1 u 2 du
du sin xdx
u 7 u 5 du
1 8 1 6
u u c
8 6
1 8 1
cos x cos 6 x c
8 6
40. 3 sin n x and cos n x
Usually done by substitution u sin x or u cos x
e.g. i cos 5 x sin 3 xdx
cos5 x1 cos 2 x sin xdx u cos x
u 5 1 u 2 du
du sin xdx
u 7 u 5 du
Both powers odd
1 8 1 6
u u c Choose either as u
8 6
Usually the higher power
1 8 1
cos x cos 6 x c
8 6
42. ii sin 6 x cos3 xdx
sin 6 x1 sin 2 x cos xdx
43. ii sin 6 x cos3 xdx
sin 6 x1 sin 2 x cos xdx u sin x
du cos xdx
44. ii sin 6 x cos3 xdx
sin 6 x1 sin 2 x cos xdx u sin x
u 6 1 u 2 du du cos xdx
u 6 u 8 du
45. ii sin 6 x cos3 xdx
sin 6 x1 sin 2 x cos xdx u sin x
u 6 1 u 2 du du cos xdx
u 6 u 8 du
1 7 1 9
u u c
7 9
1 7 1 9
sin x sin x c
7 9
46. ii sin 6 x cos3 xdx
sin 6 x1 sin 2 x cos xdx u sin x
u 6 1 u 2 du du cos xdx
u 6 u 8 du
One power odd & one power even
1 7 1 9
u u c Choose even as u
7 9
1 7 1 9
sin x sin x c
7 9
47. ii sin 6 x cos3 xdx
sin 6 x1 sin 2 x cos xdx u sin x
u 6 1 u 2 du du cos xdx
u 6 u 8 du
One power odd & one power even
1 7 1 9
u u c Choose even as u
7 9
1 7 1 9
sin x sin x c
7 9
iii sin 2 x cos 2 xdx
48. ii sin 6 x cos3 xdx
sin 6 x1 sin 2 x cos xdx u sin x
u 6 1 u 2 du du cos xdx
u 6 u 8 du
One power odd & one power even
1 7 1 9
u u c Choose even as u
7 9
1 7 1 9
sin x sin x c
7 9
iii sin 2 x cos 2 xdx sin 2 x1 sin 2 x dx
49. ii sin 6 x cos3 xdx
sin 6 x1 sin 2 x cos xdx u sin x
u 6 1 u 2 du du cos xdx
u 6 u 8 du
One power odd & one power even
1 7 1 9
u u c Choose even as u
7 9
1 7 1 9
sin x sin x c
7 9
iii sin 2 x cos 2 xdx sin 2 x1 sin 2 x dx
sin 2 x sin 4 x dx
50. ii sin 6 x cos3 xdx
sin 6 x1 sin 2 x cos xdx u sin x
u 6 1 u 2 du du cos xdx
u 6 u 8 du
One power odd & one power even
1 7 1 9
u u c Choose even as u
7 9
1 7 1 9
sin x sin x c
7 9
iii sin 2 x cos 2 xdx sin 2 x1 sin 2 x dx
sin 2 x sin 4 x dx
1 1 3 1 1
x sin 2 x x sin 2 x sin 4 x c
2 4 8 4 32
51. ii sin 6 x cos3 xdx
sin 6 x1 sin 2 x cos xdx u sin x
u 6 1 u 2 du du cos xdx
u 6 u 8 du
One power odd & one power even
1 7 1 9
u u c Choose even as u
7 9
1 7 1 9
sin x sin x c
7 9
iii sin 2 x cos 2 xdx sin 2 x1 sin 2 x dx
sin 2 x sin 4 x dx
1 1 3 1 1
x sin 2 x x sin 2 x sin 4 x c
2 4 8 4 32
1 1
x sin 4 x c
8 32
52. ii sin 6 x cos3 xdx
sin 6 x1 sin 2 x cos xdx u sin x
u 6 1 u 2 du du cos xdx
u 6 u 8 du
One power odd & one power even
1 7 1 9
u u c Choose even as u
7 9
1 7 1 9
sin x sin x c
7 9
iii sin 2 x cos 2 xdx sin 2 x1 sin 2 x dx
sin 2 x sin 4 x dx
Both powers even
1 1 3 1 1
Use sin x 1 cos x
2 2 x sin 2 x x sin 2 x sin 4 x c
2 4 8 4 32
or cos 2 x 1 sin 2 x 1 1
x sin 4 x c
8 32
54. 4 tan n x or cot n x
tan xdx log cos x c
55. 4 tan n x or cot n x
tan xdx log cos x c
tan xdx
2
56. 4 tan n x or cot n x
tan xdx log cos x c
tan xdx sec x 1dx
2 2
57. 4 tan n x or cot n x
tan xdx log cos x c
tan xdx sec x 1dx
2 2
tan x x c
58. 4 tan n x or cot n x
tan xdx log cos x c
tan xdx sec x 1dx
2 2
tan x x c
tan 3 xdx
59. 4 tan n x or cot n x
tan xdx log cos x c
tan xdx sec x 1dx
2 2
tan x x c
tan 3 xdx tan xsec 2 x 1dx
tan x sec 2 xdx tan xdx
60. 4 tan n x or cot n x
tan xdx log cos x c
tan xdx sec x 1dx
2 2
tan x x c
tan 3 xdx tan xsec 2 x 1dx
tan x sec 2 xdx tan xdx u tan x
du sec 2 xdx
61. 4 tan n x or cot n x
tan xdx log cos x c
tan xdx sec x 1dx
2 2
tan x x c
tan 3 xdx tan xsec 2 x 1dx
tan x sec 2 xdx tan xdx u tan x
udu tan xdx du sec 2 xdx
62. 4 tan n x or cot n x
tan xdx log cos x c
tan xdx sec x 1dx
2 2
tan x x c
tan 3 xdx tan xsec 2 x 1dx
tan x sec 2 xdx tan xdx u tan x
udu tan xdx du sec 2 xdx
1
u 2 log cos x c
2
1 2
tan x log cos x c
2
64. tan 4 xdx tan 2 xsec 2 x 1dx
tan 2 x sec 2 xdx tan 2 xdx
65. tan 4 xdx tan 2 xsec 2 x 1dx
tan 2 x sec 2 xdx tan 2 xdx u tan x
du sec 2 xdx
66. tan 4 xdx tan 2 xsec 2 x 1dx
tan 2 x sec 2 xdx tan 2 xdx u tan x
u du tan xdx
2 2 du sec 2 xdx
67. tan 4 xdx tan 2 xsec 2 x 1dx
tan 2 x sec 2 xdx tan 2 xdx u tan x
u du tan xdx
2 2 du sec 2 xdx
1 3
u tan x x c
3
1 3
tan x tan x x c
3
70. 5 sec n x or cosecn x
sec xsec x tan x
sec xdx
sec x tan x
dx
71. 5 sec n x or cosecn x
sec xsec x tan x
sec xdx
sec x tan x
dx
sec 2 x sec x tan x
dx
sec x tan x
72. 5 sec n x or cosecn x
sec xsec x tan x
sec xdx
sec x tan x
dx
sec 2 x sec x tan x
dx
sec x tan x
logsec x tan x c
73. 5 sec n x or cosecn x
sec xsec x tan x
sec xdx
sec x tan x
dx
sec 2 x sec x tan x
dx
sec x tan x
logsec x tan x c
sec 2 xdx tan x c
74. 5 sec n x or cosecn x
sec xsec x tan x
sec xdx
sec x tan x
dx
sec 2 x sec x tan x
dx
sec x tan x
logsec x tan x c
sec 2 xdx tan x c
sec3 xdx
75. 5 sec n x or cosecn x
sec xsec x tan x
sec xdx
sec x tan x
dx
sec 2 x sec x tan x
dx
sec x tan x
logsec x tan x c
sec 2 xdx tan x c
sec3 xdx sec x sec 2 xdx
76. 5 sec n x or cosecn x
sec xsec x tan x
sec xdx
sec x tan x
dx
sec 2 x sec x tan x
dx
sec x tan x
logsec x tan x c
sec 2 xdx tan x c
sec3 xdx sec x sec 2 xdx u sec x
77. 5 sec n x or cosecn x
sec xsec x tan x
sec xdx
sec x tan x
dx
sec 2 x sec x tan x
dx
sec x tan x
logsec x tan x c
sec 2 xdx tan x c
sec3 xdx sec x sec 2 xdx u sec x
du sec x tan xdx
78. 5 sec n x or cosecn x
sec xsec x tan x
sec xdx
sec x tan x
dx
sec 2 x sec x tan x
dx
sec x tan x
logsec x tan x c
sec 2 xdx tan x c
sec3 xdx sec x sec 2 xdx u sec x
du sec x tan xdx dv sec 2 xdx
79. 5 sec n x or cosecn x
sec xsec x tan x
sec xdx
sec x tan x
dx
sec 2 x sec x tan x
dx
sec x tan x
logsec x tan x c
sec 2 xdx tan x c
sec3 xdx sec x sec 2 xdx u sec x v tan x
du sec x tan xdx dv sec 2 xdx
80. 5 sec n x or cosecn x
sec xsec x tan x
sec xdx
sec x tan x
dx
sec 2 x sec x tan x
dx
sec x tan x
logsec x tan x c
sec 2 xdx tan x c
sec3 xdx sec x sec 2 xdx u sec x v tan x
sec x tan x sec x tan 2 xdx du sec x tan xdx dv sec 2 xdx
81. 5 sec n x or cosecn x
sec xsec x tan x
sec xdx
sec x tan x
dx
sec 2 x sec x tan x
dx
sec x tan x
logsec x tan x c
sec 2 xdx tan x c
sec3 xdx sec x sec 2 xdx u sec x v tan x
sec x tan x sec x tan 2 xdx du sec x tan xdx dv sec 2 xdx
sec x tan x sec xsec 2 x 1dx
sec x tan x sec3 xdx sec xdx
sec x tan x sec3 xdx logsec x tan x
82. 5 sec n x or cosecn x
sec xsec x tan x
sec xdx
sec x tan x
dx
sec 2 x sec x tan x
dx
sec x tan x
logsec x tan x c
sec 2 xdx tan x c
sec3 xdx sec x sec 2 xdx u sec x v tan x
sec x tan x sec x tan 2 xdx du sec x tan xdx dv sec 2 xdx
sec x tan x sec xsec 2 x 1dx
sec x tan x sec3 xdx sec xdx
sec x tan x sec3 xdx logsec x tan x
2 sec3 xdx sec x tan x logsec x tan x c
83. 5 sec n x or cosecn x
sec xsec x tan x
sec xdx
sec x tan x
dx
sec 2 x sec x tan x
dx
sec x tan x
logsec x tan x c
sec 2 xdx tan x c
sec3 xdx sec x sec 2 xdx u sec x v tan x
sec x tan x sec x tan 2 xdx du sec x tan xdx dv sec 2 xdx
sec x tan x sec xsec 2 x 1dx
sec x tan x sec3 xdx sec xdx
sec x tan x sec3 xdx logsec x tan x
2 sec3 xdx sec x tan x logsec x tan x c
1 1
sec3 xdx sec x tan x logsec x tan x c
2 2
84. 5 sec n x or cosecn x
sec xsec x tan x
sec xdx
sec x tan x
dx
sec 2 x sec x tan x
dx
sec x tan x
logsec x tan x c
sec 2 xdx tan x c
sec3 xdx sec x sec 2 xdx u sec x v tan x
sec x tan x sec x tan 2 xdx du sec x tan xdx dv sec 2 xdx
sec x tan x sec xsec 2 x 1dx
Odd powers
sec x tan x sec xdx sec xdx
3
Done by parts
sec x tan x sec3 xdx logsec x tan x
2 sec3 xdx sec x tan x logsec x tan x c
1 1
sec3 xdx sec x tan x logsec x tan x c
2 2
87. sec 4 xdx sec 2 x1 tan 2 x dx u tan x
du sec 2 xdx
88. sec 4 xdx sec 2 x1 tan 2 x dx u tan x
1 u du
2 du sec 2 xdx
89. sec 4 xdx sec 2 x1 tan 2 x dx u tan x
1 u du
2 du sec 2 xdx
1 3
u u c
3
1 3
tan x tan x c
3
90. sec 4 xdx sec 2 x1 tan 2 x dx u tan x
1 u du
2 du sec 2 xdx
1 3 Even Power
u u c
Factorise as sec xsec x
3 2 2 some power
1 3
tan x tan x c Substitute sec 2 x 1 tan 2 x
3
Use u tan x
91. sec 4 xdx sec 2 x1 tan 2 x dx u tan x
1 u du
2 du sec 2 xdx
1 3 Even Power
u u c
Factorise as sec xsec x
3 2 2 some power
1 3
tan x tan x c Substitute sec 2 x 1 tan 2 x
3
Use u tan x
Exercise 2C; 1, 2, 4, 5, 7, 8, 10, 12, 14, 16, 17, 18