Category: mathematics

Factorization of quadratic expressions
Factorization of Quadratic expression involves breaking down a mathematical expression of the form ax² + bx+ c, where a, b and c are constants and a≠0. a is called the coefficient of x², b, the coefficient of x and c the constant term.

To factorize a quadratic expression, we express it in such a way that the process of expansion can be worked backwards. This requires that the term in x be replaced by two terms chosen in such a way that the grouping method of factorization can be used.

Factorization of quadratic expressions is a key algebraic skill that helps us rewrite a quadratic expression into simpler, meaningful factors. Instead of working with an expanded form like $ax^2 + bx + c$ ax2+bx+c, factorization allows us to express it as a product of two binomials, making it easier to solve equations, find roots, and understand the behavior of graphs. This topic builds on earlier ideas of factors and multiplication and plays an important role in many areas of mathematics, including graphing parabolas and solving real-life problems involving area, motion, and optimization.

factorization of quadratic expression when coefficient x2 = 2

consider the expression:

(i)(x-3)(x+4) = x²+4x-3x-12 = x²+ x-12

(ii)(x+5)(x+1)= x² +x+5x+5 = x² +6x+5

please note: the sum of constant terms in the brackets is equal to coefficients of x in the expanded term.

that is, -3 + 4 = 1 and 5+1=6

The coefficient of x in the first expression = 1 while the coefficient of x in the second expression =6.

The product of the constant factors in the products gives the value of the constant term in the expanded expression.

in the first expression, -3 x 4 = -12 and 1 x 5 = 5

conclusion:
1. The sum of the constant terms in the factors is equal to the coefficient of x in the expression
2. the product of the constant terms in the factors is equal to the constant terms of the expression
Example 1

Factorize the expression x²+6x+8

solution

the factors will be of the form (x + a)(x + b)

a+b = 6 and a x b = 8

we need to find the values of a and b that satisfies this conditions.

one of the best way is to list the products of 8. That is:

1 x 8 = 8

2 x 4 = 8

4 x 2 = 8

we add the coefficients of the products above:

1+8 = 9

2+4 = 6

so the factors of a and b are 2 and 4. Hence the factorized form of the expression is

(x+2)(x+4)

Revision Exercise
1. Factorize the following quadratic exprerssions
(a) x² – 5x +6 (b) x²+2x-35 (c)x²-10x+24

(d)x² +4x -21 (e) x² – 5x -6

Factorizing coefficient of x2 when a > 1

consider the expression: (4x+3)(2x+1)

we expand: 4x(2x+1)+3(2x+1) = 8x²+4x+6x+3 = 8x²+10x+3

From the quadratic expressions above, comparing ax²+bx+c

we see that a=8, b=10 and c=3

we cannot use product of 3 to get factors that make b which is 10.

but we should get product of ac, such that 3×8 = 24

product of 24 includes:
- 8×3
- 12×2
- 6×4
Among this, the product that gives the sum of 10 is 6 and 4

hence the factor bx = 6x+4x=10x

hence 8x²+10x+3 = 8x²+6x+4x+3

we then group them together into two: (8x²+6x)+(4x+3)

the expression then becomes : 2x(4x+3)+1(4x+3)

hence the factorized form of 8x²+10x+3 =(2x+1)(4x+3)

Generally, to factorize the expression of the form ax²+bx+c, find two numbers such that their product is ac and their sum is b. You should then split the middle term into two terms such that their coefficients are the numbers determined.

Example problem on factorization

Factorize 4x² +6x -18

solution

The product of coefficient of x² and that of c will be 4 x -18 = -72

The factors of -72 whose sum is 6 are -6 and 12 . That is; -6+12 = 6

splitting the bx term we get: 4x² +12x-6x-18 = 4x(x+3)-6(x+3)

=(4x-6)(x+3)

note that (4x-6) can be factorized father because 2 can divide both 4 and 6.

hence we have 2(2x-3)

hence 4x²+6x-18 = 2(2x-3)(x+3)

2. Factorize the following quadratic expressions:

(a) 2x² +3x -2 (b) 3x² – 2x -8 (c) 4x² +7x +3

(d) 5x² – 21x +4 (e) 3x² +11x +6 (f) 14x² – 16x +2
1. The square of a number is 5 more than three times the number. Find the number.
2. ABC is an isosceles triangle in which AB= AC. The size of angles ABC and ACB are (3x²-2x+4)^o and (9x-6)^o respectively. Calculate the two possible sets of values of the three angles of triangle ABC.
3. A picture measuring 6cm by 9cm is mounted on a frame so as to leave a uniform margin of width x cm all around. If the area of the margin is 36cm². find x
Related topics
- Introduction to matrices
- Quadratic Equations
February 15, 2026
Introducing Quadratic expressions
Quadratic expressions are algebraic expressions of the form ax²+bx+c where a, b, and c are constants and a!=0. Constants means real numbers.

if a=0, the expression would be linear rather than quadratic. When a graph of a quadratic expression is plotted against x ,is a parabola is formed.

If a is greater than 0, the curve opens upwards as shown.

shape of a quadratic curve when a > 0

If a is less than zero, the curve opens downwards as shown.

shape of a quadratic curve when a < 0

examples of quadratic expressions is like:
- 5x² + 4x – 3
- x² + 2x – 1
- -3x² + 4x +10
Expanding algebraic expressions

Expanding algebraic expressions involves removing parentheses and simplifying the expression by multiplying terms and summing the like terms. Expansion makes use of the distributive property of algebra which states that a(b+c)=ab+ ac

consider the product of (x + y) and (a + b). The product can be written as (x + y)(a + b). if we let (a + b) = k, then we have kx + ky. this can be expanded to:

(a + b) x + (a+ b) y = ax + bx + ay + by

From the illustrations above, one can see that each term in the first bracket is multiplied by each term in the second bracket.

Example 1

Expand : (4x + 5)(4y + 10)

solution

each term in the first bracket is multiplied by every term in the second bracket. so we have:

4x(4y + 10 ) + 5 (4y + 10) = 16xy +40x + 20y +50

Example 2

Expand (6a – 3) (4b + 7)

solution

(6a – 3) (4b + 7) =6a(4b + 7) -3 (4b + 7) = 24ab + 42a – 12b – 21

Example 3

Expand:

(x+45)(y–14)

solution

(x+45)(y–14)=x(y–14)+45(y–14)

xy–14x+45y–420

Quadratic expressions

Quadratic expression is an algebraic expression where the highest power is 2. Most common quadratic equations has three terms which are usually referred to as quadratic terms. The term with a power of 2 is known as the quadratic term. The term with power of one is known as the linear term. The term with power of zero in the variable is known as the constant term.

Consider the general quadratic expression ax²+bx+c where a is NOT 0.

a is known as the coefficient of x² , b the coefficient of x and c the constant term.

consider the quadratic expression 3x²+7x +23.

3x² is the quadratic term, 7x is the linear term while 23 is the constant term.

A quadratic expression must have the quadratic term which should be the term with the highest power of the variable.

practice problem

State why each of the following is not a quadratic expression.
1. 5x-1
2. 4x³ + 7x² +3x + 15
3. 6x⁴ + 5x³ + 3x² +11x +18
Example 4

Expand and simplify each of the following:

(a) (3x – 4)(2x + 1)

(b) (2x – 7)(x-8)

(c) (3k – 2)(k-7)

solution

(a) we multiply the second bracket with each of the terms in the first bracket

(3x–4)(2x+1)=3x(2x+1)–4(2x+1)=6x²+3x–8x−4=6x²−5x−4

(b) multiplying (x-8) by 2x and -7, we obtain;

$$2x(x-8)-7(x-8)$$ $$2x^2-16x-7x+56 = 2x^2-23x+56$$

(c)

(3k–2)(k−7)=3k(k−7)−2(k−7)=3k²−21k–2k+14=3k²−23k+14

The quadratic expression identities

Quadratic identities are formulas or equalities that hold true for all values of the variable. These identities are useful for simplifying expressions and solving equations. Some of the most important quadratic identities include:
1. (a + b)²= a² + 2ab + b²
2. (a – b)²= a² – 2ab + b²
3. (a + b)(a-b) = a² – b²
we will prove each of the above identities in the next sections.

1. The Square of a Binomial in quadratic expressions

consider the expression: (a+b)(a+b)

we establish that: (a+b)(a+b) = (a+b)²= a² + 2ab + b² in all cases.

consider the square below:

The square has sides of (a + b) units. It is divided into 4 parts A, B , C and D. A is a square of side length a units while D is a square of side length b units.

B is a rectangle of length a units and width b units. C is a Rectangle of sides a and b.

The area of the whole square = (a +b)²

Area of A = a².

Area of B = ab

Area of C = ab

Area of D = b²

The total area of the square with sides (a + b) will thus be:

a² + ab + ab + b²

2. The Difference of Squares

This identity expresses the difference of two squares as a product of binomials: That is a²−b²=(a+b)(a−b)

Related topics
February 12, 2026

Introduction to Matrices

Matrices is the plural of a matrix. A matrix is a rectangular array of numbers whose value and position in the arrangement is significant.

It is common to store large quantities of data and numerical information in form of tables.

converting tables to matrices

The table below shows marks scored by three students in an exam.

Name	Mathematics	Physics	chemistry
Jane	40	59	48
Phyllis	56	63	65
James	78	81	71

the data can be written as matrices form as :

$$ A = \begin{pmatrix} 40& 59 & 48 & \\ 56 & 63 & 65 & \\ 78 & 81& 71 & \\ \end{pmatrix} $$

consider the table below that shows English soccer results. A win scores 3 points, draw 1 point while loss wins zero points.

Team	matches played	wins	Draws	loses	Goals	points
Arsenal	25	18	5	2	49	59
Man city	25	15	5	5	51	50
Aston Villa	25	14	5	6	36	47
Man U	24	13	6	5	46	45
chelsea	23	12	5	6	45	41
Liverpool	23	11	7	5	40	40
Brentford	22	11	8	3	39	41
Sunderland	22	10	9	3	27	39
Fullham	21	9	4	8	35	31
Newcastle	19	8	5	5	35	29

with time, we may get to know what each team is scoring and remember each column values. In that case, we only need to remember the patterns only. The numerical data will thus be presented in form of a matrix as shown.

$$scores = \pmatrix{25 & 18 & 5 & 2 & 49 & 59\\ 25 & 15 & 5 & 5 & 51 & 50\\ 25 & 14 & 5 & 6 & 36 & 47\\ 24 & 13 & 6 & 5 & 46 & 45\\ 23 & 12 & 5 & 6 & 45 & 41\\ 23 & 11 & 7 & 5 & 40 & 40\\ 22 & 11 & 8 & 3 & 39 & 41\\ 22 & 10 & 9 & 3 & 27 & 39\\ 21 & 9 & 4 & 8 & 35 & 31\\ 19 & 8 & 5 & 5 & 35 & 29\\ }$$

We have just converted a table to a matrix

A matix is represented as a brackets or square brackets as shown:

$$\pmatrix{ a & b & c \\ d & e & f \\ g & h & i } \ or \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} $$

Capital letters in bold are commonly used to denote a matrix.

Each number in a matrix is referred to as an element of a matrix.

Order of matrices

Matrix size and shape depends on the number of rows and columns it has. If the number of rows equals number of columns, then it is a square matrix, otherwise it is a rectangular matrix.

A matrix with m rows and n columns is said to be a matrix of order m x n .

The order of matrix denotes the number of rows and columns in the matrix.

Therefore, a matrix of order 4 x 3 will have 4 rows and 3 columns.

A matrix of order 1 x n is called a row matrix.

A matrix of order m x 1 is said to be a column matrix.

A square matrix is a matrix of order m x m or n x n.

The matrix below is a square matrix of order 4 x 4:

$$ A = \begin{pmatrix} 3 & 7 & 7 & 0 \\ 5 & 2 & 4 & 6 \\ 2 & 8 & 1 & 9 \\ 4 & 0 & 8 & 9 \end{pmatrix} $$

positioning elements in matrices

Position of an element in a matrix is described by use of suffices.

If A is a matrix and a_m,n is an element in that matrix, then a_m,n is the element in the m^th row and on the n^thcolumn.

consider the matrix above, 1 is an element at a_3,3position.

summary

practice exercise on matrices

The table below shows marked scored in three tests by grade 10 students.

Student	Test 1	Test 2	Test 3
Jane	68	73	56
Milliam	79	71	67
Nancy	67	59	63
James	45	56	78
Mark	62	70	86
Eliud	71	74	82
Brian	89	84	87
Harriet	84	76	77

(a) Write down the information from the table into a matrix and state order of the matrix.

(b) Write test 2 as a column matrix and state it’s order

(d) change the marks in test 1 and test to to 60% of the current value.

(e) convert test 3 to 40% of the current value.

(f) make a new matrix by adding a fourth column in the matrix above with the totals of (d) and (e) above

2. Given matrix A below. state the elements in:

$$A= A = \begin{pmatrix} 2 & -1 & 4 & 0 & 7 \\ 5 & 3 & -2 & 6 & 1 \\ -4 & 8 & 0 & 9 & -3 \\ 1 & 5 & 6 & -7 & 2 \end{pmatrix} $$

(a) a_1,3 (b) a_2,4 (c) a_3,2(d)a_4,1

Related topics

Databases

Practical-PHP-7-MySQL-8-and-MariaDB-Website-Databases-2nd-Edition Downloa

February 9, 2026

Operations on Matrices
1. Introduction

Matrices are rectangular arrangements of numbers organized in rows and columns. They are powerful tools used to represent and manipulate data in science, economics, engineering, computer graphics, and real life decision-making. Operations on Matrices includes addition, subtraction, multiplication and division, just like arithmetic operations on numbers.

computer graphics

Example of a Matrix
```
A = | 2   4 |
    | 1   3 |
```
- This is a 2 × 2 matrix
- 2 rows, 2 columns
2. Types of Matrices (Quick Review)

Type Description
Row Matrix Only one row
Column Matrix Only one column
Square Matrix Rows = Columns
Zero Matrix All elements are zero
Identity Matrix 1s on the main diagonal

Identity Matrix (2 × 2)
```
I = | 1   0 |
    | 0   1 |
```
Operations on Matrices

A. Addition of Matrices

Two matrices can be added only if they have the same order.

Rule:

Add corresponding elements.
```
A = | 2   3 |      B = | 1   4 |
    | 5   6 |          | 7   2 |

A + B = | 2+1   3+4 |
        | 5+7   6+2 |

      = | 3   7 |
        |12   8 |
```
📌 Real-life idea: Adding daily sales from two shops.

B. Subtraction of Matrices

Same rule as addition, but subtract corresponding elements.
```
A − B = | 2−1   3−4 |
        | 5−7   6−2 |

      = | 1  −1 |
        |−2   4 |
```
📌 Real-life idea: Comparing profits between two months.

C. Scalar Multiplication

Multiply each element by a number (scalar).
```
Let A = | 2   1 |
        | 3   4 |

3A = | 6   3 |
     | 9  12 |
```
📌 Real-life idea: Increasing production costs by a fixed rate.

D. Multiplication of Matrices

Matrix multiplication is not commutative:

AB ≠ BA (in most cases)

For matrix multiplication, the following conditions need to be met.
1. Number of columns in first matrix = number of rows in second matrix.
```
A = | 1  2 |      B = | 3 |
    | 4  5 |          | 6 |
```
```
AB = | (1×3 + 2×6) |
     | (4×3 + 5×6) |

   = | 15 |
     | 42 |
```
📌 Real-life idea: Calculating total cost using quantity × price.

4. Real-Life Applications of Matrices

1. Economics & Business

Matrices are used to calculate profit, loss, production costs, and wages.
```
Cost Matrix × Quantity Matrix = Total Cost
```
2. Computer Graphics

Matrices are used to rotate, enlarge, and shrink images in video games and animation.
```
[Image Points] × [Transformation Matrix]
```
3. Transportation

Used to organize routes, distances, and schedules between cities.

4. Science & Engineering

Matrices model electrical circuits, population growth, and chemical reactions.

5. Worked Examples

Simple Example

Add:
```
A = | 2  1 |
    | 3  4 |

B = | 5  6 |
    | 1  2 |
```
Solution:
```
A + B = | 7  7 |
        | 4  6 |
```
Complex Example

Multiply:
```
A = | 1  2  3 |
    | 4  5  6 |

B = | 1  2 |
    | 3  4 |
    | 5  6 |
```
Solution:
```
AB = | (1×1+2×3+3×5)   (1×2+2×4+3×6) |
     | (4×1+5×3+6×5)   (4×2+5×4+6×6) |

   = | 22   28 |
     | 49   64 |
```
6. Revision Exercise

(Arranged from simple to complex)

A. Basic Level
1. Add:
```
| 3  2 |   +   | 1  4 |
| 5  1 |       | 2  3 |
```
1. Subtract:
```
| 7  5 |   −   | 4  2 |
| 6  3 |       | 1  1 |
```
B. Intermediate Level
1. If
```
A = | 2  3 |
    | 1  4 |
```
Find 3A.
1. Determine whether the following multiplication is possible:
```
(2 × 3) × (2 × 2)
```
C. Advanced Level
1. Multiply:
```
| 1  0  2 |   ×   | 3 |
| 2  1  1 |       | 1 |
                  | 2 |
```
1. If
```
A = | 1  2 |
    | 3  4 |
```
Find A².

7. Application Questions (Real Life)
1. A shop sells two products. The cost matrix is
```
| 50  30 |
```
and the quantity matrix is
```
| 10 |
| 20 |
```
Find the total cost.
1. A school records boys and girls in two classes using matrices. How can matrix addition help find the total number of students?
1. A graphics designer enlarges an image using a transformation matrix. Explain how scalar multiplication is applied.
1. A transport company uses a matrix to represent distances between cities. How can matrix multiplication help calculate total travel distances?
Related topics
- Introduction to matrices
- Gradients of line
  - Matrices Basic Operations.pdf
January 27, 2026
Integration of trigonometric expressions
Integrating trigonometric expressions involves solving trigonometric expressions that contain trigonometric functions. These functions are periodical. They denote the relationship between the angle and the sides of a right-angled triangle.

Examples of trigonometric functions includes:
- sine
- cosines
- tangents
- secants
- cosecants
- cotangents
There are various techniques used to integrate mathematical expressions with trigonometric functions. Integration of trigonometric expressions requires a mix of direct integration rules. This may includes simplification using identities, substitution, and sometimes integration by parts, depending on the form of the expression. some of the techniques includes:

some know trigonometric functions and identities

some trigonometric identities are known to transform to another specific and definite trigonometric functions after the integration process. This can be very useful during Integrating trigonometric expressions. Some examples includes:

$$\int sinxdx = -cosx+c$$

The above expression shows that we always obtain a negative value of a cosine function. This negative value is accompanied by another constant number whenever the sine function is integrated.

$$\int cosxdx = sinx+c$$

in other words, integrating a cosine functions results to a positive value of a cosine function when Integrating trigonometric expressions.

(iii).∫sec²xdx=tanx+c

(iv).∫csc²xdx=−cotx+c

(v).∫secxtanxdx=−secx+c

(vi).∫cscxcotxdx=−cscx+c

2. Using trigonometric identities

by using trigonometric identities, some trigonometric expressions that could be complex to integrate becomes simplified .

When Integrating trigonometric expressions useful trigonometric identities includes:

$$sin^2x = \frac{1-cos2x}{2}$$

$$cos^2x = \frac{1+cos2x}{2}$$

$$sinxcosx = \frac{1}{2}sin2x$$

The above identities are known as power reduction identities

3. Substitution Method

Substitution method in integration is a used to simplify an integral by making a change of variables. This transforms the integral into a simpler form Integrating trigonometric expressions. It is often used when the integrand contains a composite function, making it easier to integrate.

Substitution generally has the following steps:
- Identifying part of the integrand that can be replaced with a new variable (usually denoted by u).
- Compute the derivative of the chosen substitution with respect to x (i.e., find du). This helps in expressing dx in terms of du.
- Substitute u and du into the original integral, replacing the x-terms with the new u-terms. This aims at simplifying the integral so that it’s easier to solve.
- Perform the integration with respect to u.
- Once the integral is solved in terms of u, substitute the original expression for u back into the result to return to the variable x.
In trigonometric functions, the trigonometric ratio can have even or odd powers, which determines how they are treated during integration.

Integration of trigonometric expressions: 4. Integration by Parts

Integration by parts is a technique commonly used to integrate product of two functions. It is based on the product rule for differentiation derived from the following formula:

$$\int udv = uv – \int vdu $$

By applying the formula the above formula, you can reduce the complexity of the integral so that it is easier to solve.

Integration by parts may involves steps like:
- Choose two functions from the integrand, u and dv, such that the integral of v is easier to compute than the original one
- Compute du, the derivative of u.
- Compute v, the integral of dv
- Substitute u, dv, du, and v into the integration by parts formula.
- If the resulting integral is easier to evaluate, solve it. If not, apply integration by parts again
Integration of trigonometric expressions: Trigonometric Substitution

This is used when the integrand involves square roots and trigonometric functions. consider the integral:

$$\int \frac{dx}{\sqrt{1-x^2}}$$

An integral that looks like the above expression can be solved first by substituting x for sinθ such that x -sinθ = 0.

that is: x= sinθ

Integrating Even powers of sine and cosine

Integrating even powers of sine and cosine involves simplifying the integral of expressions where the powers of sine n is an even integer. These integrals can often be simplified using power reduction identities.

When powers of a trigonometric ratios are even, the double angle identities becomes very useful. These identities reduce the even powers of sine and cosine to sums of trigonometric functions of lower powers. This process makes the integration easier. These identities includes:

$$cos^2\theta = \frac{1}{2}(1+cos2\theta)$$

$$sin^2\theta = \frac{1}{2}(1-cos2\theta)$$

Example problem on trigonometric substitution

$$cos^4d\theta = \int cos^2θcos^2θ = \int \frac{1}{2}(1+cos2θ) \frac{1}{2}(1+cos2θ)dθ$$

$$ \int \frac{1}{4}(1+cos2θ)^2dθ= \frac{1}{4}(1+cos2θ)(1+cos2θ)dθ$$

now we expand the binomial product to have:

$$ \int \frac{1}{4}(1+cos2θ)(1+cos2θ)dθ= \frac{1}{4}[1(1+cos2θ)+cos2θ(1+cos2θ)]dθ$$

$$=\frac{1}{4} \int (1+cos2θ+cos2θ+cos^22θ)dθ=\frac{1}{4} \int (1+2cos2θ+cos^22θ)dθ$$

$$cos^2 2θ = \frac{1}{4}(1+cos4θ)$$

Therefore:

$$=\frac{1}{4} \int (1+cos2θ+cos2θ+cos^22θ)dθ=\frac{1}{4} \int (1+2cos2θ+\frac{1}{4}(1+cos4θ)dθ$$

$$=\frac{1}{4}\int dθ+\frac{1}{2}\int cos2θ + \frac{1}{8}\int dθ+\frac{1}{8}\int cos4θdθ $$

$$=\frac{1}{4}\theta+\frac{1}{2}\frac{sin2\theta}{2}+\frac{1}{8}\theta+\frac{1}{8}\theta+\frac{1}{8}\frac{sin4\theta}{8} +c$$

simplifying we finally have:

$$=\frac{3θ}{8}+\frac{sin2θ}{4}+\frac{sin4θ}{32}+c$$

Example2

solve : ∫sin²4θdθ

solution

using double angle identities;

$$sin^24θ = \frac{1}{2}(1-cos8θ)dθ$$

remember the identity:

$$sin^2nθ = \frac{1}{2}(1-sin2nθ)$$

hence

$$sin^24θ = \int \frac{1}{2}(1-cos8θ)dθ$$

$$=\frac{1}{2} \int dθ – \frac{1}{2} \int cos8θdθ$$

integrating we find :

$$\frac{1}{2}θ-\frac{1}{16}sin8θ+c = \frac{θ}{2} – \frac{sin8θ}{16}+c$$

Solving Integrating trigonometric expressions with odd powers

while integrating this kind of trigonometric expressions, we use trigonometric identity which states that:

cos²θ+sin²θ=1

consider the integral:

∫sin⁵θdθ

we split the operation sin⁵θ into sin⁴θ(sinθ).

thus:

but sin⁴θ = (1-cos²θ)² from the identity stated earlier.

Expanding:

(1−cos²θ)²=(1−2cos²θ+cos⁴θ)θdθ

it follows that:

∫sin⁴θsinθdθ=∫(1−cos²θ)²sinθdθ

=∫(1−2cos2θ−cos4θ)sinθdθ

=∫(sinθ–2sinθcos²θdθ+sinθcos⁴θ)dθ

=∫sinθdθ–2∫sinθcos²θdθ+∫sinθcos⁴θdθ

remember the identity:

$$sin\theta cos^n \theta d \theta = \frac{-1}{n+1}cos^{n+1} \theta +c $$

hence

$$\int sin θcos^4θdθ = \frac{-1}{5}cos^5θ+c$$

similarly:

$$\int sinθcos^2θdθ = \frac{-1}{3}cos^3θ+c$$

hence,

$$=\int sinθdθ – 2 \int sinθcos^2θdθ+sinθcos^4θdθ $$$$= -cosθ+\frac{2}{3}cos^3θ-\frac{1}{5}cos^5θ+c$$

In general, integrating even power of sine and cosine constitutes the following procedure:
- If the integrand contains an even power of sine or cosine, use the corresponding power reduction identity. This will simplify the expression.
- After applying the identity, the integral often becomes easier to solve. This is because the powers of sine or cosine are reduced to terms involving cos⁡(2x) or constants. These terms can be integrated straightforwardly.
- Repeat where necessary: If higher powers of sine or cosine appear, you may need to apply the power reduction identity. You might have to do this multiple times.
Example 3

solve: ∫cos³θdθ

solution

$$cos^3θ = cos^2θcosθ = (1-sin^2θ)cosθdθ =(cosθ-sin^2θ)$$

hence:

$$cos^3θ dθ= cos^2θcosθdθ $$ $$= (1-sin^2θ)cosθdθ =(cosθ-sin^2θcosθ)dθ$$

$$=(cosθ-sin^θcosθ)dθ = \int cosθdθ-\int sin^2θcosθdθ$$

and

$$ \int cosθdθ-\int sin^2θcosθdθ =sinθ-\frac{1}{3}sin^3θ+c$$

Example 4

solve:

∫tan³θdθ

solution

incase of tanθ, we use the identity :

1+tan²θ=sec2θ

∫tan³θdθ=∫tan²θtanθdθ=∫(sec²θ−1)tanθdθ

∫(sec²θ−1)tanθdθ=∫(sec²θtanθ−tanθ)dθ

=∫sec²θtanθdθ−∫tanθdθ

we can use integration by substitution to solve

∫sec²θtanθdθ

let u = tanθ

$$\frac{du}{d\theta}=sec^2 \theta d \theta$$

hence du = sec²θdθ

$$dθ=\frac{du}{sec^2dθ}$$

$$\int tanθsec^2θ \frac{du}{sec^θ}=\int tanθdu =\int udu $$

and

$$\int udu = \frac{u^2}{2}+c =\frac{ tan^2θ}{2}+c$$

and

∫tanθdθ=ln|cosθ|+c

hence

$$\int sec^2θtanθdθ – \int tanθdθ = \frac{tan^2θ}{2}+ln|cosθ|+c$$

Example 4

Solve the integral ∫sec⁶2xdx

solution

∫sec⁶2xdx=∫sec⁴2xsec²xdx=∫(1+tan²x)²sec²2xdx ——(i)

$$\int sec^62xdx = \int sec^42xsec^2xdx $$$$= \int (1+tan^2x)^2sec^22xdx——–(i)$$

that is:

$$sec^22x = 1+tan^22x$$ $$sec^42x=(sec^22x)^2=(1+tan^2 2x)^2$$

Expanding the equation (i) above:

$$\int (sec^2 2x + 2tan^2 2xsec^2 2x + tan^4 2xsec^ 2x)dx$$

which results to:

$$\int sec^ 2xdx+2 \int tan^62xsec^2xdx+\int tan^4 2xsec^2 2x dx$$

consider the integral;

$$sec^2 2xdx$$

let u=2x then:

$$\frac{dx}{du}=\frac{1}{2}$$

$$dx = \frac{1}{2}du$$

Therefore: du=2dx

$$sec^2 2xdx = \frac{1}{2} \int sec^udx = \frac{1}{2}tan u+c$$ $$\frac{tan2x}{2}+c————(ii)$$

consider the expression:

$$2 \int tan^ 2x sec^ 2x dx$$

let u = tan2x

du = 2sec²2xdx

$$dx = \frac{du}{2sec^2 2x}$$

dx=du2sec22x

the expression therefore becomes:

$$2\int tan^2 2x sec^2 2x dx = 2 \int u^2 2x \frac{du}{2sec^2 2x}$$ $$\int u^2du=\frac{u^3}{3}+c = \frac{tan^3 2x}{3}+c ——–(iii)$$

Using the same method and procedure:

$$tan^4 2x sec^2 2x dx = \frac{tan^5 2x}{10}+c ——(iv)$$

combining the results of (ii), (iii) and (iv), the integral becomes;

$$\int sec^6 2x dx = \frac{tan2x}{2}+\frac{tan^3 2x}{3}+\frac{tan^5 2x}{10}+c$$

Integration of trigonometric expressions: Multiple angles identities

Multiple angle identities are trigonometric identities that express trigonometric functions of multiple angles (such as 2θ, 3θ, etc.) in terms of simpler functions of a single angle θ. These identities are helpful in simplifying expressions involving trigonometric functions of multiple angles.

Some common multiple angles includes:
- sin(A+B) +sin(A-B) = 2sinAcosB
- cos(A+B) + cos(A-B) = 2cosAcosB
- cos(A-B)-cos(A+B) = 2sinAsinB
Example 6

work out: ∫sin5θcos3θdθ

the solution to this would be:

this follows from: sin(A+B) +sin(A-B) = 2sinAcosB

$$sinAcosB =\frac{1}{2}[sin(A+B)+sin(A-B)]$$

in our case: A=5θ and B = 3θ

hence:

$$sin5θcos3θ = \frac{1}{2}[sin(5θ+3θ)+sin(5θ-3θ)]$$

$$=\frac{1}{2}[sin8θ+sin2θ]=\frac{1}{2}sin8θ+\frac{1}{2}sin2θ$$

therefore:

$$\int sin5θcos3θdθ = \frac{1}{2} \int (sin8θ+sin2θ)dθ$$

$$\int sin8θ = \frac{1}{8}cos8θ \ \ \ \ and \int sin2θ=\frac{1}{2}cos2θ$$

it follows that:

$$\frac{1}{2}\int(sin8θ+sin2θ)dθ =\frac{1}{16}cos8θ+\frac{1}{4}cos2θ+c$$

Example 2

work out: ∫sin3θcos²θdθ

you can proceed as follow:

from double angle identities:

$$cos^2θ = \frac{1+cos2θ}{2}dθ$$

therefore

$$sin3θ(\frac{1+cos2θ}{2})dθ = \frac{1}{2}\int sin3θ+\frac{1}{2} \int sin3θcos2θ$$

but

$$sin3θcos2θ = \frac{1}{2}(sin5θ+sinθ)$$

$$\frac{1}{2} \int sin3θ +\frac{1}{2}\int sin3θcos2θ$$$$=\frac{1}{2} \int sin3θdθ+\frac{1}{2} \int \frac{1}{2}(sin5θ+sinθ)dθ$$

$$\frac{1}{2}\int sin3θdθ+\frac{1}{4} \int sin5θdθ+\frac{1}{4} \int sinθdθ$$

$$-\frac{1}{6}cos3\theta-\frac{1}{20}cos5θ-\frac{1}{4}cosθ+c$$

Related Topics
October 11, 2025
Exam questions on sequence and series
Exams test your ability to identify arithmetic sequences by looking for a constant common difference between terms, then use formulas to find any nth term (a_n) or the sum (S_n) of a series. You will be expected to apply formulas like

a_n=a₁+(n−1)d and S_n=n/2(a₁+a_n) where 𝑎₁is the first term,. d is the common difference, while n is the number of terms.

here are the questions involving sequence and series

Question 1

(a) The first term of an arithmetic progression is 3 and the sum of its first 8 terms is 164.

(i) Find the common difference of the arithmetic progression. (2 marks)
(ii) Given that the sum of the first 𝑛 terms of this arithmetic progression is 570, find 𝑛. (3 marks)
(b) The first, fifth and seventh terms of another arithmetic sequence forms a decreasing geometric progression. If the
first term of the geometric progression is 64, find
(i) The values of the common difference of the arithmetic sequence (3 marks)
(ii) The sum of the first 12 terms of the geometric progression. (2 marks)

Question 2

2. An arithmetic progression of 41 terms is such that the sum of the first five terms is 560

and the sum of the last five terms is -250. Find:

(a) The first term and the common difference (5mks)

(b) The last term (2mks)

(c) The sum of the progression (3mks)

Question 3

The first term of arithmetic progression (A.P) is a and the common difference d. the 7^th term of the A.P is 11. The sum of the first 12 consecutive terms of the A.P is 123.

(a) (i) Form two simplified equations involving a and d to represent the information (2mks)

(ii) Find the values of a and d. (3mks)

(a) The 1st, 3rd and 8th terms of the A.P from the first 3 consecutive terms of a geometric progression (G.P). Calculate;

(i) The 6th term of the G.P. (3mks)

(ii) The sum of the first 6 terms of the G.P. (2mks)

Related topics
- Introduction to arithmetic series
- sequence of numbers
October 6, 2025
Angle properties of a circle
Angle properties of a circle refer to the specific geometric rules governing the relationship between angles formed by points, arcs, chords, and tangents within a circle.

Angles subtended by an arc or a chord

The figure below helps us study the angles subtended by an arc or a chord in the same segment at the circumference.

angle a, b and c are equal. The three angles are subtended at the circumference of the circle by the chord AC. All the three angles are in the same segment AFEDC we conclude that:

Angles subtended by the same arc or chord at the circumference, in the same arc are equal.

Angle subtended by equal arcs

Equal arcs are enclosed by equal chords. If chords of a circle are equal, then they are subtends equal angles at the circumference.

In the figure below chord AC and ZY are equal. Therefore angle ABC and angle ZXY are equal.

Related topics
- The circle
October 4, 2025
The properties of a circle
The properties of a circle include its center (the fixed point), radius , diameter, and circumference. A circle is a line which curves and joins up with itself such that any point on the line is at equal distance from a fixed point.

,

properties of a circle: Parts of a circle

Circumference of a circle

Circumference is the length of the curved line forming the circle.

An arc

An arc is a fraction of the circumference made from an incomplete circle.

chord

A chord is any straight line joining two points on the circumference.

Diameter

Diameter is any chord passing through the center of a circle. Diameter is the longest chord that can be drawn in a circle.

segment

A segment is any region enclosed by a chord and an arc. A chord usually divides a complete circle has into two segments. One segment is smaller than the other.

The smaller segment is referred to as the minor segment while the larger one is referred to as the major segment.

When a chord inside the circle makes two equal chords, then the chords are called semi-circle. In other words, a semicircle is a figure created from an area enclosed by an arc and diameter of a circle.

Sector of a circle

Sector of a circle is a two dimensional figure resulting from an area being enclosed by two radii and an arc .

Any pair of radii that joins at an angle inside a circle makes two sectors. the smaller sector is called the minor sector while the bigger sector is known as the major sector.

Angle properties of a circle

Angle subtended by an arc or a chord

Consider the figure below:

The minor arc ABC subtends and angle ADC at the circumference of the circle. Angle ADC is opposite to arc ABC.

If points A and C are joined by a straight line, chord AC or arc ABC subtends angle ADC at the circumference.

consider the figure below:

The same minor arc as describe above subtends angle AOC at the centre of the circle.

Chords AC or arc ABC subtends angle AOC at the center.

The two diagrams can help us investigate properties of angles subtended by an arc or a chord in the same segment at the circumference.

we conclude that:

Angle subtended by the same arc or chord at the circumference, in the same segment are equal

Equal chords or arcs subtends equal angles at the circumference.

Related topics
- sequence of numbers
- Uniform circular motion
October 1, 2025
The sequence in numbers
A sequence in numbers is an ordered list of numbers (called terms) that follow a specific pattern or rule, such as adding, subtracting, multiplying, or dividing by a certain value to get from one term to the next. For example :
1. A sequence of counting numbers: 1,2, 3, 4, 5,….
2. Sequence of odd numbers: 1, 3, 5, 7, 9, 11,……..
3. Sequence of square numbers: 1, 4, 9, 16, 25,…….
difference between pattern and sequence

A pattern is the underlying rule or arrangement that dictates the order of items, while a sequence is the specific, ordered list of numbers or elements that follows that pattern. In other words; a pattern is the the rule(how), and a sequence is the the list of terms from the rule. in other words:

The pattern is the rule or principle governing the arrangement.

The sequence is the actual list of items that adhere to that rule.

What is a pattern?

A sequence is a list of numbers or elements arranged in a specific, connected order that follows a pattern.

In nature, pattern is the concrete, ordered set of items that results from a pattern.

A pattern can be like “add 4 to the previous number starting from 7”

The sequence would be 7, 11, 15, 19, 23, 27, 31,35,,,,,,.

Explaining sequence

A sequence is a list of numbers developed from a rule. Each number in a sequence is called a term. A sequence must come from a rule. However, a sequence can have more than one rule.

In a sequence, the terms occurs in a particular order. There is first term, second term, third term….and to infinity.

We need to establish the relationship between the value of a term and it’s position in the sequence. A position is usually represented by n. Therefore, we have 1^st term, 2^nd term, 3^rd term,………….,n^-2th term, n^-1th term, n^th term.

When nth term is known, any other term can be obtained using the formula developed from the rule.

consider the sequence with nth term given as 3n+2. This means the rule is that we multiply the position by 3 then add 2 to get the term in that position. To get a value for any position, we just substitute n for that position . As an example, we consider the first few positions for the sequence we defined above:

1st term: 3(1)+2 = 5

2nd term: 3(2) + 2 = 8

3rd term: 3(3) + 2 = 11

4th term: 3(4) + 2 = 14

5th term: 3(5) + 2 = 17

some commonly known sequences
1. Fibonacci sequence
It is a sequence obtained by adding the first two preceding’s terms to get the next term. That is:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610…..

for example, the sixth term of the sequence is the sum of 3 and 5 which are the terms preceding it. similarly 9th term is the sum of 13 and 21 which are 8th and seventh term consecutively.

example application of Fibonacci sequence

Arithmetic sequence

It is a sequence in which any two consecutive terms differs by the same number. The number is known as the common difference.

Geometric sequence

It is a sequence in which the ratio between any two consecutive terms is a constant value. For example 3,9,27,81,243…..

the ratio between 234 and 81 is 3. The ratio between 81 and 27 is 3 and so on.

Related topics
- measuring errors
- calculus
September 23, 2025
Integrating products of trigonometric functions
Products of trigonometric functions usually refers to the integral of a product of trigonometric functions over a specified range. It can be interpreted as finding area under the curve made by a trigonometric function. For example the function:

$$\int sin(x)cos(x)dx$$

The equation represents an area under the curve formed by the product of sine and cosine functions. The integral results to

$$\int -cos(x)sin(x)dx$$

in a geometrical terms, finding the integral of products of trigonometric functions can be described as finding an area under the curve defined by those functions over a certain interval. for example:

$$\int_{0}^{\pi} sin(x)cos(x)dx = -\frac{1}{4}cos(2x)+C$$

represents the area between the product of sin(x) and cos (x) functions and the x-axis from x =0 to x=π.

The result of the above integral is 0.25

The topic of integrals trigonometric products is important because these integrals are useful in areas like:
- Fourier series analysis
- modelling of wave interference patterns in physics
- describing mechanical vibrations and oscillations
- used to describe AC circuits and their oscillatory behaviors
Trigonometric functions of sine and cosines

products of sines and cosines are of the form:

$$\int sin^m(x)cos^n(x)dx$$

Integrating products trigonometric functions involves use of trigonometric identities like double angle identities depending on the form of the integral.

case 1

At least one of the indices m and n is an odd positive integer. If m is an odd positive integer , then we isolate the one one sine.

consider the identity:

$$sin^2x = 1-cos^2x$$

we express the remaining sin^m x as sin^(m-1) x and then express it in terms of cos x. That is:

$$\int sin^m(x)cos^n(x)dx = \int sin^{m-1}(x)sin(x)cos^n(x)dx$$

As an example, consider the following expression.

$$\int sin^3(x)cos^2(x)dx = \int sin^2(x)sin(x)cos^2(x)dx = \int(1-cos^2x)cos^2xsinxdx$$

$$=\int sinxcos^2xdx – \int sinxcos^4xdx = -frac{1}{3}cos^3x + \frac{1}{5}cos^5x+c$$

case 2

If both m and n are non-negative even integer, we use the half angle formula which states:

$$sin^2 \theta = \frac{1}{2}(1-cos2 \theta)$$

$$xcos^2 \theta = \frac{1}{2}(1+cos2 \theta)$$

As as an example, consider the integral:

$$sin(x)cos^2(x)dx = \int \frac{1}{2}(1-cos2x)\frac{1}{2}(1+cos2x)dx$$

please note:

(1−cos2x)(1+cos2x)=1−cos2x+cos²x–cos²2x=1−cos²2x

therefore:

$$\frac{1}{4}(1-cos^22x)dx = \frac{1}{4} \int (1-\frac{1}{2}(1+cos4x)dx$$

$$\frac{1}{4} \int (1-\frac{1}{2}(1+cos4x)dx=\frac{1}{4}dx – \frac{1}{8}dx-\frac{1}{8}cos4xdx$$

14∫(1−12(1+cos4x)dx=14∫dx–18∫dx–18∫cos4xdx=14x–18x–133sin4x+c=18(x−14sin4x)+c

$$\frac{1}{4}x – \frac{1}{8}x – \frac{1}{32}sin4x + c = \frac{1}{8}(x-\frac{1}{4}sin4x)+c $$

Related Topics
September 14, 2025