How to Solve Anti Derivatives

In mathematics, anti derivative is also called as integration process. Integration is an important concept in mathematics and, together with differentiation, is one of the two main operations in calculus. The term integral may also refer to the notion of antiderivative, a function F whose derivative is the given function f. In this case it is called an indefinite integral. Integral can be classified as definite and indefinite integral. (Source: Wikipedia)

General formula for integration:

∫ `(x^n)` = `(x^(n + 1) / (n + 1))` + c

Example problems for how to solve anti derivatives

Anti derivative example problem 1:

Solve:

Integrate the given function ∫ (14.3x4 + 31.7x3 - 6x) dx.

Solution:

Given ∫ (14.3x4 + 31.7x3 - 6x) dx

Integrate the given function with respect to x, we get

∫ (14.3x4 + 31.7x3 - 6x) dx = ∫ 14.3x4 dx + ∫ 31.7x3 dx - ∫ 6x dx.

= 14.3 (x5 / 5) + 31.7 (x4 / 4) - 6 (x2 / 2) + c.

= `(14.3 / 5)` x5 + `(31.7 / 4)` x4 - 3 x2 + c.

Answer:

The final answer is  `(14.3 / 5)` x5 + `(31.7 / 4)` x4 - 3 x2 + c.

Anti derivative example problem 2`:`

Solve:

Find the value of the integration

`int_0^3(x^10)dx`

Solution:

Integrate the given function with respect to x, we get

`int_0^3(x^10)dx`  = `(x^11 / 11)`30

Substitute the lower and upper limits, we get

= `((3^11 / 11) - (0^11/ 11))`

= `((177147 / 11) - (0 / 11))`

= `(177147 / 11)`

Answer:

The final answer is `(177147 / 11)`

Anti derivative example problem 3:

Solve:

Integration using algebraic rational function ∫ `(15 dx) / (8x + 21)`

Solution:

Using integrable function method,

Given function is ∫ `(15dx) / (8x + 21)`

Formula:

∫ [L / (ax + c)] dx = (L / a) log (ax + c)

From given, L = 15, a = 8, and c = 21

Integrate the given equation with respect to x, we get

= `(15 / 8)` log (8x + 21)

Answer:

The final answer is `(15 / 8)` log (8x + 21).

Practice problems for how to solve anti derivatives

Anti derivative practice problem 1:

Solve:

Integrate the given function using integrable function ∫ `(2 / (12x + 103))` dx

Answer:

The final answer is `(1 / 6)` log (12x +103)

Anti derivative practice problem 2:

Solve:

Integrate the given function using integrable function ∫ `(27 / (21x + 48))` dx

Answer:

The final answer is `(9 / 7)` log (21x + 48)

Algebra is widely used in day to day activities watch out for my forthcoming posts on 3x3 Matrix Multiplication and cbse class 10. I am sure they will be helpful.

Anti derivative practice problem 3:

Solve:

Integrate the given function ∫ (27.39x3 - 17.132x) dx

Answer:

The final answer is `(27.39 / 4)` x4 - `(17.132 / 2)` x2 + c

Reading and Decimal Writing

The numbers left to the decimal point is known as whole number and right to the decimal point is called decimal number. The number in the digit has place value. Depending upon the place value we can read and write the decimals.

For example: 3456.89 Here 3456 is the whole numbers and .89 is the decimal numbers. Let us see about place value of the decimal used for read and write the decimal.

Understanding Decimal Symbol is always challenging for me but thanks to all math help websites to help me out. 

Place Value – Reading and Decimal Writing:

Let take the example decimals 35926.94843 denotes the place value of the following table.

Place value	Decimals Number
Ten thousands	3
Thousands	5
Hundreds	9
Tens	2
ones	6
Decimal point	.
Tenths	9
Hundredths	4
Thousandths	8
Ten thousandths	4
Hundred thousandths	3

Example Problems – Reading and Decimal Writing:

Example 1:
Write the place value of the simple decimals of this 6978.4254
Solution:
6 x 1000 thousands
9 x 100 hundreds
7 x 10 tens
8 x 1 ones
4 x 0.1 tenths
2 x 0.01 hundredths
5 x 0.001 thousandths
4 x 0.0001 ten-thousandths

Example 2:
Convert this mixed fraction of 39 49/100 to decimals and how to read and the decimal form.

Solution:

Mixed Number	Expanded Form		Writing Decimal form	Reading  the decimal (phrase form)
89 `(43)/(100)`	(8 * 10) + 9 * 1	4* (`(1)/(10)` )+  3 * (`(1)/(100)`)	89.43	Thirty nine and four tenths nine hundredths
	Whole number part	Fractional part

Example 3:
How to read and write the following fraction to decimal form?
  `(74)/(100)`, `(87)/(1000)` and `(65)/(10)`

Solution:

Fraction  Number	Reading decimal		Writing Decimal form
`(74)/(100)`	Seven tenths	Four hundredth	0.74
`(87)/(1000)`	Eight hundredth	Seven thousandth	0.087
`(65)/(10)`	Six ones	Five tenth	6.5

Practice Problem – Reading and Decimal Writing:

1.  How to read the decimal form 33. 67?
Answer: Thirty three and sixty seven hundredth
2. Write in decimal form of the phrase seven hundred fifty three and four hundred and forty three thousandths.
Answer: 753.443

Median of Three Numbers

In statistics, the median is the middle value of the set of data after the arrangement of ascending or descending order. If the total of elements in a set is even, median is arithmetic mean of two middle numbers (n and n+1). Divide total number of elements by 2 to get n. In this lesson we will discuss about how to find the median of three numbers.

Median of Three Numbers – Example Problems

Example 1: In a class 3 students’ marks as follows 76, 71, and 65. Find the median of given data.

Solution:

Arrange the numbers in ascending order {65, 71, 76}

Median:

{65, 71, 76}

Total number of elements in a data set is 3 which is an odd number. So find the middle value which is the median.

Here, 71 is the middle value.

Therefore 71 is the median.

Example 2: Find the median of three numbers 69, 72, and 87.

Solution:

Arrange the numbers in ascending order {69, 72, 87}

Median:

{69, 72, 87}

Total number of elements in a data set is 3 which is an odd number. So find the middle value.

Here 72 is the middle value.

Therefore median of given three numbers is 72.

Example 3: Three students’ height in a class as follows 158, 178, and 184. What is the median?

Solution:

Arrange the numbers in ascending order {158, 178, 184}.

Median:

{158, 178, 184}

Total number of elements in a data set is 3 which is an odd number. So find the middle value.

Here, 178 is the middle value.

Therefore 178 is the median.

Example 4: Three students’ weight in a class as follows 88, 75, and 84. What is the median?

Solution:

Arrange the numbers in ascending order {75, 84, 88}.

Median:

{75, 84, 88}


Total number of elements in a data set is 3 which is an odd number. So find the middle value.

Here, 84 is the middle value.

Therefore 84 is the median.

Median of Three Numbers – Practice Problems

Problem 1: Find the median of three numbers 145, 78, and 256.

Problem 2: Three students’ height in a class as follows 192, 178, and 182. Find the median.

Answer: 1) 145 2) 178

Binomial Nomenclature List

Introduction to Binomial nomenclature

The species concept: Species is recognized as the smallest unit of taxonomy or the basic unit of classification. In simpler form every different and recognizable type of organism (animal, plant or microbe) is a different species. In scientific terms, a species could be defined as a group of closely resembling organisms of the same type, in which members can interbreed freely and naturally to produce viable and fertile offspring, whereas they cannot interbreed freely or produce viable and fertile offspring with members outside the group.

Binomial nomenclature

Binomial nomenclature is the scientific method of giving names to every species of living organism. Carolos Linnaeus first introduced it in 1758. In this method every species of organism is given a name consisting of two words. The first word would refer to the name of the genus to which the organism belongs in the existing system of classification. Therefore this is known as the generic name. The second word in the name is called the specific name, since it refers to the exact species to which the organism belongs.

Rules to be followed in giving scientific names to plants and animals on the basis of binomial nomenclature

• The name must consist of two words both in Latin and in other languages in Latinized form.
• The first word should refer to the genus to which the organism belongs and it should start with capital letter.
• The second word should refer to the species to which the organism belongs and it starts with a small letter.
• Names derived by binomial nomenclature should also include reference in abbreviation to the name this is known as author citation.
• The generic and specific names when printed must be in italics and when written otherwise must be separately underlined, indicating their Latin origin. Author citation is not in italics nor is it underlined.
• When the same organism is given more than one scientific name by different authors, the earlier name becomes accepted.
• These rules are framed and presented as International Code for botanical Nomenclature (ICBN) for plants and as International Code for Zoological Nomenclature (ICZN) for animals. They are periodically reviewed and revised by specific committees.
  
  

Advantages of binomial nomenclature

• Names derived from binomial nomenclature gives universal reorganization to the organism.
• Scientific name avoid confusion arising out of the same organism having different common names in different origins of the world.e.g. For brinjal in India (Solanum melongena) is called egg plant in western countries.
• Using binomial nomenclature system automatically establishes taxonomic interrelationships between different taxa.
• Addition of new information, acquiring rare information, exchange of resource materials, and exchange of information etc., become easy when universally accepted scientific names are used.

Geometry Solid Figures

Geometry solid figures are a branch of geometry which mainly focuses on the properties like surface area, and volume of solid figures or three dimensional shapes like Rectangular Prism, cube, cone, cylinder, and sphere. Geometry solid figures generated by the revolution of a two dimensional plane. For example a cylinder is generating by the revolution of a rectangle; the revolution of a circle about its diameter generates a sphere.

Formulas for geometry solid figures:-

Geometry solid figures of Cube:-


Surface area = 6a2 square units, Volume = a3 cubic units.

Geometry solid figures of Cylinder:-


Lateral surface area = 2πrh square units, Total surface area = 2πr (h + r) square units, Volume = πr2h cubic units

Geometry solid figures of Cone:-


Lateral surface area = πrs square units, Total surface area = πrs + πr2 square units, Volume = 1/3 πr2h cubic units

Geometry solid figures of Sphere:-


Surface area of a sphere = 4πr2 square units, Volume = 4/3 πr3 cubic units

Geometry solid figures of Rectangular Prism:-


The equation for find the volume (V) of a rectangular prism is V = wdh.

Example geometry solid figures problems:-

Example problem1:-

Find the volume of cylinder given the radius is 6 cm and 11 cm.

Solution:-

Volume of cylinder = r2 h cubic units.

= (3.14) * 62 * 11

= 3.14 * 36 * 12

= 1 356.48cm3

Example problem2:-

Find the volume of cone given the radius is 4 cm and 8 cm.

Solution:-

Volume of cone = 1/3 r2 h cubic units.

= 1/3 (3.14) * 42 * 8

= 0.33 * 3.14 * 16 * 8

= 132.6336cm3

Example problem3:-

Find the volume of cube with the side length of 8 cm.

Solution:-

Volume of cube = a3

= 83

= 512 cm3

Example problem4:-

A rectangular prism contains at width for 3 inches and depth 6 inches and height 4 inches then finds the volume of rectangular prism?

The faces are known as width (w), depth (d), and height (h).

Solution:-

Volume (V) of a rectangular prism is:

V = wdh.

Now we can calculate,

V= wdh

V= 3 in x 6 in x 4 in

V = 72 in3

Example problem5:-

The sphere has the radius of 4m.find the surface area of the sphere.

Solution:-

Radius (r) = 4 m

Surface area of the sphere = 4 π r2 square unit

= 4 x 3.14 x (4)2

=4 x 3.14 x 16

= 200

Surface area of the sphere =200 m2.

How to Reduce Binomials

Binomial is also an algebraic equation but it has only two terms. (a+b) It’s  a example of binomial. And the binomial theorem commonly used to expand the binomials to any given power without direct multiplication .Here we are going to learn about how to reducing binomials. Let us see about binomial with examples.

More information about binomials

The following binomial shows, how we can write the binomials using powers
(a + b)⁰ is equal to  1
(a + b)¹ can be written as  a + b
(a + b)² can be written as  a² + 2ab + b²
(a + b)³ can be written as a³ + 3a²b + 3ab² + b³
(a + b)⁴ can be written as  a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴
(a + b)⁵ can be written as  a⁵ + 5a⁴b + 10a³b² + 10a²b³ + 5ab⁴ + b⁵

Example problems for how to reduce binomials:

Example 1:
Reduce  `((x+2)^2)/(x+2)`
Solution:

Step 1: We can write it as an .`((x+2)^2)/(x+2)`

Step 2: And the question can be written as  `((x+2)(x+2))/(x+2)`

Step 3: On both the numerator and denominator, the (x+2) will be cancelled.

Step 4: Therefore, the answer is (x+2).

Example 2:
Reduce 2ab+5yz-ab-3yz+9ab.
Solution:

Step 1: First we need to arrange the terms.

Step 2: So, 2ab-ab+9ab+5yz-3yz.

Step 3: Here, we need to add the terms.

Step 4: Therefore the answer is 10ab+2yz.

Example 3:
Reduce  `((x+7) (x+9))/((x+81)^2 (x+49)^2)`
Solution:

Step 1: It can be written as  `((x+7) (x+9))/((x+81)^2 (x+49)^2)`

Step 2: And, we can write the given question is `((x+7)(x+9))/((x+9)(x+9)(x+7)(x+7))`

Step 3: And Here both the sides the (x+7) and (x+9 ) are canceled.

Step 4: Therefore the answer  is`(1)/((x+9)(x+7))`

Example 4:
Reduce 4ts+6gh-ts-2gh+7ts.
Solution:

Step 1: First we need to arrange the terms.

Step 2: So, 4ts+7ts-ts+6gh-2gh.

Step 3: Here, we need to add the terms.

Step 4: Therefore the answer is 10ts+4gh

Example 5:
Reduce:  `((a+b)^3 (5a))/((a+b) (25))`
Solution:

Step 1: The question can be written as `((a+b)(a+b)(a+b)(5a))/((a+b) (25))`

Step 2: On both the numerator and denominator, the (a+b) and 5 will be canceled.

Step 3: Therefore, the answer is `((a+b)^2 (a))/(5)`

These are the examples of how to reducing binomials.

Scale Factor Triangle

A scale area is a digit to scales otherwise multiplies, a few number. In the equation y=Cx, C represents the scale factor x. C is as well as represents the coefficient of x with might be recognized the stable of proportionality of y toward x. For illustration, doubling-up distance communicate to a scale factor of 2 meant for distance, as unkind a block in half consequences in part by a scale factor of ½.


Definition of scale factor triangle:

The ratio of any two corresponding lengths in two similar geometric figures is called as Scale Factor.
The ratio of the length of the scale drawing to the corresponding length of the actual object is

called as Scale Factor.   (source:wikipedia)


More about Scale Factor:

Scale factor triangle is an item that is applies to calculate the size of the entity. The conditions scale factor is utilize to declare the proportions of the geometric form. In geometry, the expression scale factor is frequently used to modify the proportions of the image. Some shapes aspect knows how to be improved otherwise reduce. In online, scholar knows how to study regarding different subject. In online, scholar is able to study on scale factor obviously. Online knowledge is appealing with interactive. Furthermore online knowledge is dissimilar as of group area education in dissimilar method. In online, scholar contain single to ones knowledge.

A scale factor triangle is a form applies as a form through that a further shape is develops in scale.

A scale factor is innovative to scale form in 1 toward 3 sizes.

Scale factor know how to be recognized through the subsequent situation

Examples problem for scale factor triangle:

Example 1:

find the image point p(3,4) order dilation with center(0,0) with scale factor of 3.

Solution:

Step1:   To find the point on the coordinate plane under dilation with center as origin multiply the coordinates with scale     factor.

Step 2:   The points of P(3,4) is P'(3*3, 4*3)

Step 3:   The value is P'(9,12)


Example 2:

find the image point p(2,6) order dilation with center(0,0) with scale factor of 2.

Solution:

Step1:   To find the point on the coordinate plane under dilation with center as origin multiply the coordinates with scale factor.

Step 2:   The points of P(2,6) is P'(2*2, 6*2)

Step 3:   The value is P'(4,12)