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.
Weekly Verde Valley Real Estate Transaction ReportDamian Bruno
This document provides a summary of residential real estate listings in Yavapai County, Arizona as of December 27, 2010. It includes listings for homes in areas such as Clarkdale, Bridgeport, Camp Verde, Cornville/Page Springs, Cottonwood, and Verde Village. Details are provided for several homes listed as active, active-contingent removal, and pending including addresses, listing prices, features like number of beds and baths, lot sizes, and brief descriptions. Contact information is also included for the listing agents.
The document discusses various transformations that can be applied to a graph y=f(x) to generate other graphs. These transformations include: shifting the graph up or down by adding a constant a to y, shifting left or right by adding a to x, reflecting the graph across the x-axis or y-axis, reflecting parts of the graph where f(x)<0 or x>0, stretching the graph vertically or horizontally by multiplying y or x by a constant k, and combining multiple transformations.
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.
11X1 T05 06 line through point of intersection (2010)Nigel Simmons
The document provides steps to find the equation of a line passing through the intersection of two other lines and a given point. It first finds the intersection point of the two initial lines as (-2,3). It then calculates the slope of the line between this point and the given point (1,2) as -3. Finally, it derives the equation of the line as 7x + 21y - 49 = 0.
The document discusses the point slope formula and provides examples of its use. Specifically, it contains:
1) The point slope formula: y - y1 = m(x - x1)
2) An example that finds the equation of the line passing through (-3,4) and (2,-6).
3) A second example that finds the equation (3x + 4y + 6 = 0) of the line passing through (2,-3) that is parallel to the given line (3x + 4y - 5 = 0).
The document discusses reduction formulae, which expresses a given integral as the sum of a function and a known integral. It provides an example of using integration by parts to derive the reduction formula for the integral ∫cosnxdx. This formula expresses the integral In in terms of the integral In-2. The document then applies this formula to evaluate the integral ∫cos5xdx. It also derives the reduction formula for the integral ∫cotnxdx and states that this formula can be used to find the value of I6.
The document discusses the formula for calculating the perpendicular distance from a point to a line. It states that the perpendicular distance is the shortest distance. The formula is given as d = (Ax1 + By1 + C)/√(A2 + B2), where (x1, y1) are the coordinates of the point and Ax + By + C = 0 is the equation of the line. An example is worked through to find the equation of a circle given its tangent line and center point. It also discusses how the sign of (Ax1 + By1 + C) indicates which side of the line a point lies.
All straight lines can be written in the form of y = mx + b, where m is the slope and b is the y-intercept. Alternatively, lines can be written in the general form of Ax + By + C = 0, where A, B, and C are integers or surds. Lines parallel to the x-axis have the form y = c, where c is a constant. Lines parallel to the y-axis have the form x = k, where k is a constant. The example shows finding the equation of a line perpendicular to another line in general form.
Media Services charges $40 for a phone and $30 per month for its economy plan. A linear model can be used to determine the total cost C(t) of operating the phone for t months, where C(t) = $40 + $30t. Using this model, the total cost for 3 months of service is $40 + $30 * 3 = $40 + $90 = $130.
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
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.
Weekly Verde Valley Real Estate Transaction ReportDamian Bruno
This document provides a summary of residential real estate listings in Yavapai County, Arizona as of December 27, 2010. It includes listings for homes in areas such as Clarkdale, Bridgeport, Camp Verde, Cornville/Page Springs, Cottonwood, and Verde Village. Details are provided for several homes listed as active, active-contingent removal, and pending including addresses, listing prices, features like number of beds and baths, lot sizes, and brief descriptions. Contact information is also included for the listing agents.
The document discusses various transformations that can be applied to a graph y=f(x) to generate other graphs. These transformations include: shifting the graph up or down by adding a constant a to y, shifting left or right by adding a to x, reflecting the graph across the x-axis or y-axis, reflecting parts of the graph where f(x)<0 or x>0, stretching the graph vertically or horizontally by multiplying y or x by a constant k, and combining multiple transformations.
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.
11X1 T05 06 line through point of intersection (2010)Nigel Simmons
The document provides steps to find the equation of a line passing through the intersection of two other lines and a given point. It first finds the intersection point of the two initial lines as (-2,3). It then calculates the slope of the line between this point and the given point (1,2) as -3. Finally, it derives the equation of the line as 7x + 21y - 49 = 0.
The document discusses the point slope formula and provides examples of its use. Specifically, it contains:
1) The point slope formula: y - y1 = m(x - x1)
2) An example that finds the equation of the line passing through (-3,4) and (2,-6).
3) A second example that finds the equation (3x + 4y + 6 = 0) of the line passing through (2,-3) that is parallel to the given line (3x + 4y - 5 = 0).
The document discusses reduction formulae, which expresses a given integral as the sum of a function and a known integral. It provides an example of using integration by parts to derive the reduction formula for the integral ∫cosnxdx. This formula expresses the integral In in terms of the integral In-2. The document then applies this formula to evaluate the integral ∫cos5xdx. It also derives the reduction formula for the integral ∫cotnxdx and states that this formula can be used to find the value of I6.
The document discusses the formula for calculating the perpendicular distance from a point to a line. It states that the perpendicular distance is the shortest distance. The formula is given as d = (Ax1 + By1 + C)/√(A2 + B2), where (x1, y1) are the coordinates of the point and Ax + By + C = 0 is the equation of the line. An example is worked through to find the equation of a circle given its tangent line and center point. It also discusses how the sign of (Ax1 + By1 + C) indicates which side of the line a point lies.
All straight lines can be written in the form of y = mx + b, where m is the slope and b is the y-intercept. Alternatively, lines can be written in the general form of Ax + By + C = 0, where A, B, and C are integers or surds. Lines parallel to the x-axis have the form y = c, where c is a constant. Lines parallel to the y-axis have the form x = k, where k is a constant. The example shows finding the equation of a line perpendicular to another line in general form.
Media Services charges $40 for a phone and $30 per month for its economy plan. A linear model can be used to determine the total cost C(t) of operating the phone for t months, where C(t) = $40 + $30t. Using this model, the total cost for 3 months of service is $40 + $30 * 3 = $40 + $90 = $130.
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
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 nth roots of unity. It states that the solutions to equations of the form zn = ±1 are the nth roots of unity. These solutions form a regular n-sided polygon with vertices on the unit circle when placed on an Argand diagram. As an example, it shows that the solutions to z5 = 1 are the fifth roots of unity located at angles that are integer multiples of 2π/5 around the unit circle. It then proves that if ω is a root of z5 - 1 = 0, then ω2, ω3, ω4 and ω5 are also roots. Finally, it proves that 1 + ω + ω2 + ω3 + ω4 = 0.
1. Integration By
Partial Fractions
Ax
To find; dx
P x
2. Integration By
Partial Fractions
Ax
To find; dx
P x
1 If degA x degP x , perform a division
3. Integration By
Partial Fractions
Ax
To find; dx
P x
1 If degA x degP x , perform a division
2 If degA x degP x , factorise P x
4. Integration By
Partial Fractions
Ax
To find; dx
P x
1 If degA x degP x , perform a division
2 If degA x degP x , factorise P x
A
a) for linear factor x a , write
xa
5. Integration By
Partial Fractions
Ax
To find; dx
P x
1 If degA x degP x , perform a division
2 If degA x degP x , factorise P x
A
a) for linear factor x a , write
xa
b) for multiple linear factors x a , write
n
A B C
x a x a 2
x a n
6. Integration By
Partial Fractions
Ax
To find; dx
P x
1 If degA x degP x , perform a division
2 If degA x degP x , factorise P x
A
a) for linear factor x a , write
xa
b) for multiple linear factors x a , write
n
A B C
x a x a 2
x a n
Ax B
c) for polynomial factors e.g. ax bx c, write 2
2
ax bx c
8. x2 x
e.g. i dx x 1 x2 0x 0
x 1
x2 x
9. x2 x 1
e.g. i dx x 1 x2 0x 0
x 1
x2 x
x0
10. x2 x 1
e.g. i dx x 1 x2 0x 0
x 1
x2 x
x0
x 1
11. x2 x 1
e.g. i dx x 1 x2 0x 0
x 1
x2 x
x0
x 1
1
12. x2 x 1
e.g. i dx x 1 x2 0x 0
x 1
x 1 1 dx x2 x
x 1
x0
x 1
1
13. x2 x 1
e.g. i dx x 1 x2 0x 0
x 1
x 1 1 dx x2 x
x 1
x0
1 x 1
x 2 x log x 1 c
2 1
14. x2 x 1
e.g. i dx x 1 x2 0x 0
x 1
x 1 1 dx x2 x
x 1
x0
1 x 1
x 2 x log x 1 c
2 1
3dx
ii
x2 x
15. x2 x 1
e.g. i dx x 1 x2 0x 0
x 1
x 1 1 dx x2 x
x 1
x0
1 x 1
x 2 x log x 1 c
2 1
3dx
ii
x2 x
3dx
x x 1
16. x2 x 1
e.g. i dx x 1 x2 0x 0
x 1
x 1 1 dx x2 x
x 1
x0
1 x 1
x 2 x log x 1 c
2 1
3dx A B 3
ii 2
x x x x 1 x x 1
3dx
x x 1
17. x2 x 1
e.g. i dx x 1 x2 0x 0
x 1
x 1 1 dx x2 x
x 1
x0
1 x 1
x 2 x log x 1 c
2 1
3dx A B 3
ii
x2 x x x 1 x x 1
3dx A x 1 Bx 3
x x 1
18. x2 x 1
e.g. i dx x 1 x2 0x 0
x 1
x 1 1 dx x2 x
x 1
x0
1 x 1
x 2 x log x 1 c
2 1
3dx A B 3
ii
x2 x x x 1 x x 1
3dx A x 1 Bx 3
x x 1
x0
A3
A 3
19. x2 x 1
e.g. i dx x 1 x2 0x 0
x 1
x 1 1 dx x2 x
x 1
x0
1 x 1
x 2 x log x 1 c
2 1
3dx A B 3
ii
x2 x x x 1 x x 1
3dx A x 1 Bx 3
x x 1
x0 x 1
A3 B3
A 3
20. x2 x 1
e.g. i dx x 1 x2 0x 0
x 1
x 1 1 dx x2 x
x 1
x0
1 x 1
x 2 x log x 1 c
2 1
3dx A B 3
ii
x2 x x x 1 x x 1
3dx A x 1 Bx 3
x x 1
3 3 x0 x 1
dx A3 B3
x x 1
A 3
21. x2 x 1
e.g. i dx x 1 x2 0x 0
x 1
x 1 1 dx x2 x
x 1
x0
1 x 1
x 2 x log x 1 c
2 1
3dx A B 3
ii
x2 x x x 1 x x 1
3dx A x 1 Bx 3
x x 1
3 3 x0 x 1
dx A3 B3
x x 1
3 log x 3 log x 1 c A 3
22. x2 x 1
e.g. i dx x 1 x2 0x 0
x 1
x 1 1 dx x2 x
x 1
x0
1 x 1
x 2 x log x 1 c
2 1
3dx A B 3
ii
x2 x x x 1 x x 1
3dx A x 1 Bx 3
x x 1
3 3 x0 x 1
dx A3 B3
x x 1
3 log x 3 log x 1 c A 3
x 1 c
3 log
x
24. x5
iii 2 dx
x 3 x 10
x5
dx
x 5 x 2
25. x5 x5
iii 2 dx A
B
x 3 x 10 x 5 x 2 x 5 x 2
x5
dx A x 2 B x 5 x 5
x 5 x 2
26. x5 x5
iii 2 dx A
B
x 3 x 10 x 5 x 2 x 5 x 2
x5
dx A x 2 B x 5 x 5
x 5 x 2 x 2
7B 3
3
B
7
27. x5 x5
iii 2 dx A
B
x 3 x 10 x 5 x 2 x 5 x 2
x5
dx A x 2 B x 5 x 5
x 5 x 2 x 2 x5
7B 3 7 A 10
3 10
B A
7 7
28. x5 x5
iii 2 dx A
B
x 3 x 10 x 5 x 2 x 5 x 2
x5
dx A x 2 B x 5 x 5
x 5 x 2 x 2 x5
10 3 7B 3 7 A 10
dx
7 x 5 7 x 2 3 10
B A
7 7
29. x5 x5
iii 2 dx A
B
x 3 x 10 x 5 x 2 x 5 x 2
x5
dx A x 2 B x 5 x 5
x 5 x 2 x 2 x5
10 3 7B 3 7 A 10
dx
7 x 5 7 x 2 3 10
10 3 B A
log x 5 log x 2 c 7 7
7 7
30. x5 x5
iii 2 dx A
B
x 3 x 10 x 5 x 2 x 5 x 2
x5
dx A x 2 B x 5 x 5
x 5 x 2 x 2 x5
10 3 7B 3 7 A 10
dx
7 x 5 7 x 2 3 10
10 3 B A
log x 5 log x 2 c 7 7
7 7
dx
iv 3
x x
31. x5 x5
iii 2 dx A
B
x 3 x 10 x 5 x 2 x 5 x 2
x5
dx A x 2 B x 5 x 5
x 5 x 2 x 2 x5
10 3 7B 3 7 A 10
dx
7 x 5 7 x 2 3 10
10 3 B A
log x 5 log x 2 c 7 7
7 7
dx
iv 3
x x
dx
xx 2 1
32. x5 x5
iii 2 dx A
B
x 3 x 10 x 5 x 2 x 5 x 2
x5
dx A x 2 B x 5 x 5
x 5 x 2 x 2 x5
10 3 7B 3 7 A 10
dx
7 x 5 7 x 2 3 10
10 3 B A
log x 5 log x 2 c 7 7
7 7
A Bx C 1
iv 3
dx 2
x x x x 1 xx 2 1
dx Ax 2 1 Bx C x 1
xx 2 1
33. x5 x5
iii 2 dx A
B
x 3 x 10 x 5 x 2 x 5 x 2
x5
dx A x 2 B x 5 x 5
x 5 x 2 x 2 x5
10 3 7B 3 7 A 10
dx
7 x 5 7 x 2 3 10
10 3 B A
log x 5 log x 2 c 7 7
7 7
A Bx C 1
iv 3
dx 2
x x x x 1 xx 2 1
dx Ax 2 1 Bx C x 1
xx 2 1
x0
A 1
34. x5 x5
iii 2 dx A
B
x 3 x 10 x 5 x 2 x 5 x 2
x5
dx A x 2 B x 5 x 5
x 5 x 2 x 2 x5
10 3 7B 3 7 A 10
dx
7 x 5 7 x 2 3 10
10 3 B A
log x 5 log x 2 c 7 7
7 7
A Bx C 1
iv 3
dx 2
x x x x 1 xx 2 1
dx Ax 2 1 Bx C x 1
xx 2 1 xi
x0
A 1 B Ci 1
35. x5 x5
iii 2 dx A
B
x 3 x 10 x 5 x 2 x 5 x 2
x5
dx A x 2 B x 5 x 5
x 5 x 2 x 2 x5
10 3 7B 3 7 A 10
dx
7 x 5 7 x 2 3 10
10 3 B A
log x 5 log x 2 c 7 7
7 7
A Bx C 1
iv 3
dx 2
x x x x 1 xx 2 1
dx Ax 2 1 Bx C x 1
xx 2 1 xi
x0
A 1 B Ci 1
B 1 C 0
36. x5 x5
iii 2 dx A
B
x 3 x 10 x 5 x 2 x 5 x 2
x5
dx A x 2 B x 5 x 5
x 5 x 2 x 2 x5
10 3 7B 3 7 A 10
dx
7 x 5 7 x 2 3 10
10 3 B A
log x 5 log x 2 c 7 7
7 7
A Bx C 1
iv 3
dx 2
x x x x 1 xx 2 1
dx Ax 2 1 Bx C x 1
xx 2 1 xi
x0
1 x dx
A 1 B Ci 1
x x 1
2
B 1 C 0
37. x5 x5
iii 2 dx A
B
x 3 x 10 x 5 x 2 x 5 x 2
x5
dx A x 2 B x 5 x 5
x 5 x 2 x 2 x5
10 3 7B 3 7 A 10
dx
7 x 5 7 x 2 3 10
10 3 B A
log x 5 log x 2 c 7 7
7 7
A Bx C 1
iv 3
dx 2
x x x x 1 xx 2 1
dx Ax 2 1 Bx C x 1
xx 2 1 xi
x0
1 x dx
A 1 B Ci 1
x x 1
2
B 1 C 0
log x logx 2 1 c
1
2
39. v
xdx A B Cx D x
2
x 12 x 2 1 x 1 x 12 x 1 x 12 x 2 1
A x 1x 2 1 B x 2 1 Cx D x 1 x
2
40. v
xdx A B Cx D x
2
x 12 x 2 1 x 1 x 12 x 1 x 12 x 2 1
A x 1x 2 1 B x 2 1 Cx D x 1 x
2
x 1
2 B 1
1
B
2
41. v
xdx A B Cx D x
2
x 12 x 2 1 x 1 x 12 x 1 x 12 x 2 1
A x 1x 2 1 B x 2 1 Cx D x 1 x
2
x 1 xi
2 B 1 2C 2 Di i
1
B
2
42. v
xdx A B Cx D x
2
x 12 x 2 1 x 1 x 12 x 1 x 12 x 2 1
A x 1x 2 1 B x 2 1 Cx D x 1 x
2
x 1 xi
2 B 1 2C 2 Di i
1
1 C 0 D
B 2
2
43. v
xdx A B Cx D x
2
x 12 x 2 1 x 1 x 12 x 1 x 12 x 2 1
A x 1x 2 1 B x 2 1 Cx D x 1 x
2
x 1 xi
2 B 1 2C 2 Di i
1
1 C 0 D
B 2
2
x0
2A B D 0
1 1
2A 0
2 2
A0
44. v
xdx A B Cx D x
2
x 12 x 2 1 x 1 x 12 x 1 x 12 x 2 1
A x 1x 2 1 B x 2 1 Cx D x 1 x
2
1 x 1 xi
1
2 x 1 2x 1
2 2 dx 2 B 1 2C 2 Di i
1
1 C 0 D
B 2
2
x0
2A B D 0
1 1
2A 0
2 2
A0
45. v
xdx A B Cx D x
2
x 12 x 2 1 x 1 x 12 x 1 x 12 x 2 1
A x 1x 2 1 B x 2 1 Cx D x 1 x
2
1 x 1 xi
1
2 x 1 2x 1
2 2 dx 2 B 1 2C 2 Di i
1
1 C 0 D
1 1 B 2
x 1 2
2
2 x 1 dx 2
x0
2A B D 0
1 1
2A 0
2 2
A0
46. v
xdx A B Cx D x
2
x 12 x 2 1 x 1 x 12 x 1 x 12 x 2 1
A x 1x 2 1 B x 2 1 Cx D x 1 x
2
1 x 1 xi
1
2 x 1 2x 1
2 2 dx 2 B 1 2C 2 Di i
1
1 C 0 D
1 1 B 2
x 1 2
2
2 x 1 dx 2
x0
1 x 1
1
1 2A B D 0
tan x c
2 1 1 1
2A 0
2 2
A0
47. v
xdx A B Cx D x
2
x 12 x 2 1 x 1 x 12 x 1 x 12 x 2 1
A x 1x 2 1 B x 2 1 Cx D x 1 x
2
1 x 1 xi
1
2 x 1 2x 1
2 2 dx 2 B 1 2C 2 Di i
1
1 C 0 D
1 1 B 2
x 1 2
2
2 x 1 dx 2
x0
1 x 1
1
1 2A B D 0
tan x c
2 1 1 1
2A 0
1 1 2 2
tan 1 x c A0
2 x 1
48. v
xdx A B Cx D x
2
x 12 x 2 1 x 1 x 12 x 1 x 12 x 2 1
A x 1x 2 1 B x 2 1 Cx D x 1 x
2
1 x 1 xi
1
2 x 1 2x 1
2 2 dx 2 B 1 2C 2 Di i
1
1 C 0 D
1 1 B 2
x 1 2
2
2 x 1 dx 2
x0
1 x 1
1
1 2A B D 0
tan x c
2 1 1 1
2A 0
1 1 Exercise 2G; 2 2
tan 1 x c 1, 3, 5, 7 to 21 A0
2 x 1
