Trigonometric Integrals

Trigonometry is a fundamental concept of mathematics. It is used in calculus functions and vectors. In this topic we have to use trigonometry as integral function. That means how to integrate trigonometric functions. For this we also have to know what is integration?  Integration means to calculate area of a given curve, and the curve is a closed curve made by x axis and y axis.

Trigonometric integrals mean integration of trigonometric functions. As we know these trigonometric functions are basic formulas for solving trigonometric integral. To more simplify this term, let’s take an example like sin2X. This is a trigonometric function. And we integrate this function for this first we have to expand this term by using formula of trigonometry. After expanding we carry out the constant term then by using product rule of integral, we can integrate this trigonometric function.

Above example is simple it has only one trigonometric function but trigonometric function may be combine with other function also. It can be algebraic function with trigonometry, logarithmic function with trigonometry and exponential function with trigonometry. These are also called integrals of trigonometric functions. To solve this type of problem either we can use integration by substitution method or integration by parts method.

Inverse trigonometric integrals such as sin^-1X and cos ^-1X etc. now to integrate this type of functions we have to use basics of calculus. We need  to take this function equal to any constant like Y. means we have to write Y= sin ^-1X. now we transfer sin function to other site the we get. X=sin Y. Now we can simply integrate this term.

Trigonometric substitution integrals, here we also integrate trigonometric functions and calculus functions, but procedure is different. To integrate this type of function we have to substitute and equal trigonometric term in place of other trigonometric term. The first from of integrals is integration of [f’(x)/f(x)] dx=logf(x) . In this form integral of a function whose numerator is the exact derivative of its denominator and equal to the logarithmic of its denominator? The second form is, in the integrand consist of the product of a constant power of a function f(x) and the derivative of f(x), to obtain the integral we increase the index by unity and then divide by increase index. This procedure is known as power formula. Lets take an example suppose we have to integrate (4x^3/1+x^4) dx= ln (1+x^4). By using this method we substitute 1+x^4 = any constant term like (t), and after that we integrate this function.

Exponential Function an Introduction

An Exponential Function is a function which involves exponent which is the variable part rather than the base as in any normal function. For instance f(x)= x^3 is a function and an exponential function is something like g(x)= 3^x, here the exponent or the power is a variable (x) and the fixed value is the base (3). So, the definition of Exponential functions can be given as a function whose base is a fixed value and the exponent a variable. Example: f(x) = 5^x, here the base 5 is fixed value and the exponent ‘x’ is the variable.
In general, we can define Exponential Functions as a function which is written in the form ‘a^x’ in which ‘a’ is the base which is a fixed value or constant (‘a ‘not equal to 1) and ‘x’ the variable which is any real number. The most common exponential function we come across in math is e^x which is known as the Euler’s number.
Let us now take a quick look at the Exponential Function Properties. Consider the Exponential function f(x) = b^x for which the properties are as follows:
• The domain of the exponential function consist of all real numbers
• The range is the collection of all positive real numbers
• When b is greater than 1 then the function is an increasing function also called exponential growth function and when b is less than 1 then the function is a decreasing function also called exponential decay function
• The other properties that an exponential function satisfy are,
1. b^x.b^y = b^(x+y) [when bases are same and a multiplication operation then we can add the powers]
2. b^x/b^y = b^(x-y)[when bases are same and a division operation then we can subtract the powers]
3. (b^x)^y = b^(xy) [when a base is raised to a power x and raised to whole power y then we can multiply the powers]
4. a^x.b^x= (a.b)^x [when bases are different with the same power and a multiplication operation then we can multiply the bases whole raised to power]

We come across a function called an Inverse Exponential Function; this is nothing but a logarithm function.  We know that the exponential function is written in the form f(x) = b^x, to find the inverse of a given function we need to interchange x and y and solve for y. By interchanging we get x = b^y  and then solving for y gives us y = log x (base b) which is a logarithm function.

Points and lines tutoring

Tutor is the person who teaches the kids and this teaching section is the tutoring. Tutoring is an open source for the students to gain knowledge that is in online a point is nothing but the dot , it has no dimension or no width, it’s only a simple black dot. In geometry co ordinates of a point which shows the particular place in a segment for representation.Line has two end points is called segment. Line segment is denoted with a connected piece of line.line segments names  has two endpoints and it is named by its endpoints.

Points and Lines Tutoring:

Tutoring about the geometric points and lines we have to know the classification of a points and lines. points and lines classification are as follows.

Collinear points:
When three or more points lies on the same line is said to be collinear points.

Midpoint:
A halfway point where line segment divides into two equal parts are called midpoint.

Equidistant point:
A point which is said to be equidistant in a line segment where point is equal length from other points which are in congruent then the point is equidistant point.

Parallel line segment:
Two lines which does not touch each other are called parallel lines.

Perpendicular line segment:
Two line segment  that form a L shape are called perpendicular lines.

Problems in Points and Lines Tutoring:

Example 1:
Find the distance between the points A(6,3) and B (2,1).

Solution:
Let assume "d" be the distance between A and B.           (x1,y1)= (6,3), (x2,y2)= (2,1).

Then d (A, B) =`sqrt((x_2-x_1)^2+(y_2-y_1)^2)`

= `sqrt((2-6)^2 +(1-3))^2)`

= `sqrt((-4)^2+(-2)^2)`

= `sqrt(16+4)`

=`sqrt20`
=2`sqrt5`

Example 2:
Find co-ordinate of the mid point of the line segment joining given points A(-5,3) and B(2,1)

Solution:
The required mid point is
Formul a   `((x_1+x_2)/2 ,(y_1+y_2)/2)` here,  (x1, y1) = (-5,3),(x2, y2) = (2,1)

=  `((-5+2)/(2))``((3+1)/(2)) `

=   `(-3/2) ` , ` (4/2)`

=    `(-3/2, 2)`

Example 3:
Find the slope of the lines given (8,-5) and (4,2)

Solution:
(x1,y1)= (8,-5), (x2,y2)= (4,2).
We know to find slope of line,m=` (y_2-y_1) /(x_2-x_1)`

=`(2+5)/(4-8)`

m =`7/-4`

Example 4:
Find the equation of the line having slope  3 and y-intercept 5.

Solution:
Applying the equation of the line is y = mx + c
Given,       m =3 ,c = 5
y =  3x +5

or  y = 3x+5
or  -3x+y-5 = 0
3x-y+5 = 0.

Introduction for division math facts

Division is essentially the opposite of multiplication. Division finds the quotient of two numbers, the dividend divided by the divisor. Any dividend divided by zero is undefined. For positive numbers, if the dividend is larger than the divisor, the quotient is greater than one; otherwise it is less than one (a similar rule applies for negative numbers). The quotient multiplied by the divisor always yields the dividend.

Division is neither commutative nor associative. As it is helpful to look at subtraction as addition, it is helpful to look at division as multiplication of the dividend times the reciprocal of the divisor, that is a ÷ b = a × 1⁄b. When written as a product, it obeys all the properties of multiplication. (Source: Wikipedia)

Definition and Steps for Division Math Facts:

Definition for division:

Division is defined as an arithmetic function, which is the opposed process of multiplication. From the process of division, the proportion or ratio of two numbers be capable of be calculated.

Otherwise, the process of decision how many periods of one number is included in a further one. Symbol of division is ‘/’ or ‘÷’.

If we divide a number by another number, then

Dividend = (Divisor * Quotient) + Remainder

Steps for division math facts:

Step1. Division of two integers by the related signs resolve be positive sign

a) Positive integer ÷ positive integer = positive integer

b) Negative integer ÷ negative integer = positive integer

Step2. Division of two integers by the unlike signs will be negative

a) Positive integer ÷ negative integer = negative integer

b) Negative integer ÷ positive integer = negative integer.              

Division Math Facts Example Problems and Practice Problem:

Example problem for division math facts:

1. Solve the following division

36 ÷ 6

Solution:

36 ÷ 6

= (6 * 6) ÷ 6

Answer: 6

2. Solve the following division,

32 ÷ 4

Solution:

32 ÷ 4

= (4 * 8) ÷ 4

Answer: 8

3. Solve the following division,

48 ÷ 8

Solution:

48 ÷ 8

= (6 * 8) ÷ 8

Answer: 6

Practice problem for division math facts:

1.      - 49 ÷ 7 = -7 (unlike sign)

2.      56 ÷ 7 = 8 (like sign)

3.      48 ÷ 12 = 4 (like sign)

4.      81 ÷ 9 = 9(like sign).

Learning subtraction of square roots

A square of a number a is a number x. therefore x2=a .A number x whose square is a. Every positive real number a has a unique positive square root, called the principal square root. Square root denoted by a radical sign as sqrt of a. For positive ax, the principal square root can also be written in exponent notation, as a1/2. We can undo a exponent with a radical, and a radical can undo a power. The “`sqrt(a)` “symbol is called the "radical “symbol..The line across the top is called the vinculum.

Subtracting Square Root Terms

Subtracting square roots

Subtracting square roots is combining like terms when we need to do that with algebraic expressions. The induces (a square roots index is 2 `root(2)(a)` , a cube roots index is 3  `root(3)(a)` , a 4th roots index is 4 `root(4)(a)` ,a 5th roots index is 5 `root(5)(a)` etc.) or the radicands (enclosed by parentheses after SQRT or the expression under the root sign) are the same.

Just as with "regular" numbers, square roots can be subtracted together. But you might not be able to simplify the subtraction all the way down to one number. Just as "you can't subtract apples and oranges", so also you cannot combine "unlike" radicals. To subtract radical terms together, they have to have the same radical part.

Simplifying Square-Root Terms

Simplify a square root, we take out anything that is a perfect square; that is, we take out front anything that has two copies of the same factor.

We can raise numbers to powers other than just 2; we can cube things, raise them to the fourth power, raise them to the 100th power, and so forth.

(ab)^2=a^2b^2  and`sqrt(ab)`   = `sqrt(a)``sqrt(b)` but we can’t write this subtraction of square root  `sqrt(a-b)`  = `sqrt(a)` - `sqrt(b)`

Example Problems on Subtracting Square Root

Example Problems

1. (4 * `sqrt(2)` ) - (5 * `sqrt(2)` ) + (12 * `sqrt(2)` )
Solution: Combine like
= (4 - 5 + 12) * `sqrt(2)`
Answer is 11 * `sqrt(2)`

2. (53 * `sqrt(5)` ) - (5 * `sqrt(5)` )
Solution: Combine like terms
(53 - 5) * `sqrt(5)`
= 48 * `sqrt(5)`

3. (40 * `sqrt(5)` ) - (48* `sqrt(5)` )
Solution: Combine like terms by subtracting the numerical coefficients.
(40 - 48) * `sqrt(5)`
= -8 *`sqrt(5)`
`sqrt(3x+1)`-( -`sqrt(2x-1)` )= 1    subtracting 2 square roots with variables
`sqrt(3x+1)` = 1 - `sqrt(2x-1)`
Take square both sides
3x + 1 = 1 - 2 `sqrt(2x-1)` + 2x-1
3x + 1 - 1 -2x + 1 = -2 `sqrt(2x-1)`
x+1 = -2 `sqrt(2x-1)`
take Square both sides again
x^2 + 2x + 1 = 4(2x -1)
x^2 + 2x + 1 = 8x -4
x^2 -6x + 5 = 0
(x-5)(x-1) = 0
x1 = 5, x^2 = 1