tag:blogger.com,1999:blog-90771508323472307772019-06-14T10:11:01.330-07:00E-Pandu.Come-pandu.com is a media blog that provides statistical guides, mathematics about technology and bloggingE-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.comBlogger81125tag:blogger.com,1999:blog-9077150832347230777.post-34179956468990589322019-06-14T00:06:00.002-07:002019-06-14T00:06:39.808-07:00Integrals of Exponential and Logarithmic Functions
To obtain the integral
formula for exponent and logarithmic functions, let us consider the derivative
of functions presented in the following table,
Recalling that integral is
anti-differential, then:
Example 2.
Solve the integral results
of ∫ 2 ex
dx.
Answer:
∫ 2 ex dx = 2ex
+ c
Example 3.
Solve the integral results of ∫ 2 ex dx.
Answer:
∫ 2 ex dx = 2ex + c
Example 4E-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com0tag:blogger.com,1999:blog-9077150832347230777.post-50172851662536420992019-06-13T21:20:00.002-07:002019-06-13T21:20:53.179-07:00Differentiation of Exponential and Logarithmic Functions
The exponential function
derivative is not easily solved directly with the definition of the derived
function. However, it can be solved using the inverse function derivative,
namely the logarithmic function. So, in this article the derivative of the
logarithmic function will take precedence before obtaining the derivative
function derivative.
Derivatives of Logarithmic Functions
For example,E-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com0tag:blogger.com,1999:blog-9077150832347230777.post-7029498607538032222019-06-10T23:20:00.004-07:002019-06-10T23:20:35.300-07:00Intergral Partial Fraction
With the previous method
we cannot (will be difficult) to neutralize the forms as below
∫ 5x (2x – 4) dx or ∫ 2x sin x dx.
To solve the form of the equation, let us consider the following steps.
If f and g are differential functions, the rule applies.
If we integrate the rule, we will get it
To further simplify the form we can write it with
u = f(x) du = f ’E-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com0tag:blogger.com,1999:blog-9077150832347230777.post-10025139616587369362019-06-10T23:00:00.000-07:002019-06-10T23:00:41.386-07:00Use Trigonometric Substitution To Solve The Integral Of Square Root
In this article, we will
discuss how to solve integrals that contain the form of square root as follows,
These forms can be changed by substituting functions that contain trigonometry. The basic concept is to change the square root into a simple form of trigonometric function, i.e.
(A)
For Example:
(B)
For Example:
(C)
For Example:
Example 1.
Find the E-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com0tag:blogger.com,1999:blog-9077150832347230777.post-22166568815085355652019-06-09T19:59:00.000-07:002019-06-09T19:59:34.327-07:00Integration by Substitution
In this article, an
integration technique called substitution method will be discussed. The basic
concept of this method is to change integral problems into simpler forms.
The general form of
internal substitution is as follows,
Example 1.
Solve the integral results
of ∫ (x2 + 3)10 2x dx
Answer:
Example 2.
Solve the integral results of ∫ (x4 – x2)3
(8x3 – 4x) dx
Answer:
E-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com0tag:blogger.com,1999:blog-9077150832347230777.post-34510887865418522242019-06-09T19:27:00.001-07:002019-06-09T19:27:23.157-07:00How to use the chain rule with trigonometric functions
Derivatives of Trigonometric Functions
In the previous article,
we discussed the derivative of the trigonometric function expressed by the
following formulas,
f(x)
f’(x)
sin x
cos x
cos x
-sin x
By using chain theorems,
we can develop derivatives of complex trigonometric functions, such as the
following examples,
Example 1
Find the result of the E-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com0tag:blogger.com,1999:blog-9077150832347230777.post-42295591591203821982019-06-09T18:56:00.000-07:002019-06-09T18:56:20.387-07:00Chain Rule For Finding Derivatives
To solve
the derivative of function f which is f(x) = (2x + 3)2, the way you
have used so far is to describe the form first into the sum of terms, where the
result is f(x) = 4x2 + 12x + 9. What if we are asked to solve a
derivative of a function such as f(x) = (x3 + x2 -4x + 5)5
?. Even though we can describe such a form, it is clearly not effective. So to
solve such derivative multiple E-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com0tag:blogger.com,1999:blog-9077150832347230777.post-16248602393043526532019-06-09T18:20:00.001-07:002019-06-09T18:20:45.069-07:00Volume by Rotation Using Integration
In our daily lives, many
of us encounter rotating objects such as flower vases, lamp shades, buckets,
cans and so on. A question disturbs us, can we calculate the volume of the
object. It's not easy, but with the Integral method we will learn below, it can
answer that question.
Look at the buildings
below and the swivel object produced when the buildings are rotated around the
X-axis by 3600 (E-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com0tag:blogger.com,1999:blog-9077150832347230777.post-49781060712915389962019-05-23T14:41:00.001-07:002019-05-23T14:41:20.275-07:00Calculation Of The Area Between The Curve And The X Axis
After you have skillfully
calculated certain integrals, below will be presented a calculation of the area
between the curve and the X-axis.
Example 7.
Calculate the area bounded by the curve y = x2, X axis, line x = 1, and line x = 2.
Answers:
So, the area is 2.33 units of area
The size of the area shown in Figure-1 is positive, because the product of f (x) and ∂x is negative.
E-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com0tag:blogger.com,1999:blog-9077150832347230777.post-74023201153961753652019-05-17T18:11:00.000-07:002019-05-17T18:11:06.313-07:00Calculate the area of an object using the integral method
A problem that we often
face is if we are faced with the calculation of the building area that is
limited not by a straight line. An example is how we calculate the area of a
wake in Figure 1 below,
To find the building area approach is to make plots in the form of a unit square as shown in Figure 1. If we consider the area of the inner and outer areas of the building, the building areaE-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com0tag:blogger.com,1999:blog-9077150832347230777.post-50029700085516463152019-05-17T17:05:00.001-07:002019-05-17T17:05:04.805-07:00Definition of Integral
A derivative of a function
expressed with 3x2 + 7x + 5 is 6x + 7. If it is reversed, can you
determine the formula of a function if it is known that the function of the
decline is 6x + 7? The process of determining a function if the derivative is
known is what is called Anti Differential or Integral. So, integral is the
opposite of differential. To determine the integral of a function, it is notE-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com0tag:blogger.com,1999:blog-9077150832347230777.post-52905518965337543852019-05-17T16:23:00.000-07:002019-05-17T16:23:13.973-07:00Solve The Polynomial Equation By A Divisor Of The Square Form
The division of a polynomial equation for this time
only contains a divider in a linear form, namely ax + b. Next we will discuss
the division of the polynomial with a square divider or ax2 + bx + c. The
following is given an example of a polynomial division model.
Example 1.
(3x3 – 7x2 – 11x + 4) : (x2
– x – 2)
So the model of the division of the polynomial equation is
The first E-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com0tag:blogger.com,1999:blog-9077150832347230777.post-17500488106689442632019-05-17T16:01:00.004-07:002019-05-17T16:01:56.868-07:00Solving Polynomial by Factoring Equations
When you solve quadratic equations, one way to solve
it is by factoring the form. The principle used is the following
multiplication.
If a . b = 0 then a = 0 or b = 0
Similarly, for the following polynomial equations,,
untuk persamaan polinomial berikut ini,
anxn
+ an-1xn-1 + an-2xn-2 + … + a2x2
+ a1x + a0 = 0
The principle used is not different from the principle in solving E-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com0tag:blogger.com,1999:blog-9077150832347230777.post-52610482750583266172019-05-17T15:43:00.003-07:002019-05-17T15:43:57.064-07:00Remaining theorem of polynomial if divided by x-k
Suppose the value of the polynomial f(k) = 0, then the
polynomial f(x) remains 0 if divided by x - k. Thus, (x - k) is said to be a
factor of f(x).
Polynomial factor theorem
The polynomial f(x) has a factor (x - k) if and only
if f(k) = 0
Example 1.
Show that x - 3 is a factor of the polynomial f(x) = x3 – x2 – x – 15
Answers:
Apparently, the result is f(x) = 0, so x - 3 is a E-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com0tag:blogger.com,1999:blog-9077150832347230777.post-27991436479331338292019-05-11T20:49:00.000-07:002019-05-11T20:49:15.199-07:00Use Remainder Theorem to solve polynomial equations
In division numbers, we often get things like the
following example. At division 17: 5 is 3 with the remaining 2.
17 = (5 x 3) + 2
The numbers are 17 as divided numbers, 5 as dividers, 3 as divides, and 2 as the remainder.
By looking at the similarities above, we can also determine the basic equation that connects the polynomial f(x) which in this case is a divided element. P(x) as a E-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com0tag:blogger.com,1999:blog-9077150832347230777.post-44600791781285605762019-05-11T20:00:00.001-07:002019-05-11T20:00:23.284-07:00Polynomial Division
By using a method similar to dividing a number, we can
also divide the polynomials, for example (3x3 – 7x2 – 11x + 4) : (x –
4) in the following distribution model,
Information that must be known:
(i) x - 4 as a divider
(ii) 3x2 + 5x + 9 is the result of the division of the polynomial
(iii) 40 is the remainder of the division
Look again at the division model above. If the E-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com0tag:blogger.com,1999:blog-9077150832347230777.post-64195845122758456082019-05-11T19:48:00.004-07:002019-05-11T19:48:46.908-07:00Definition of Polynomials
polynomials in x degree n are a form
anxn + an-1 xn-1
+ an-2 xn-2+ … + + a2 x2 + a1x
+ a0
for n a chunk number, and a0, a1, …, an as a constant and an ≠ 0.
In a polynomial where a1, …, an is called the tribal coefficient x. The number n is called the polynomial quality. The form of writing a polynomial is arranged with the highest rank base placed in the front row, while the smaller rank E-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com0tag:blogger.com,1999:blog-9077150832347230777.post-47607390300207402552019-05-01T07:04:00.003-07:002019-05-01T07:04:46.210-07:00Solve Trigonometry equations with cos x^0 + b sin x^0 = c
To
solve the equation in the form of a cos x0 + b sin x0 = c
first, what must be done is to change the form of the equation a a cos x0 + b sin x0
= c to
be,
k cos (x - α)0 = c
considering
– 1 ≤ cos (x - α) ≤ 1, so
that the equation can be solved must be fulfilled.
Example 1.
Determine
the set of completion equations cos x0 – sin x0 = -1, for 0 ≤ x < 360.
cos x0 – sin x0 =
-1
√2 E-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com0tag:blogger.com,1999:blog-9077150832347230777.post-8161230895545646112019-05-01T07:02:00.001-07:002019-05-01T07:03:22.117-07:00Solve Trigonometry equations with a cos x^0 + b sin x^0
Change the form a cos x0 + b
sin x0 to be k cos (x – a)
With the summation formula that has been
learned in school like
cos (a – b) = cos a cos b + sian a sin b
we can declare form 4 cos (x – 30)0
to form
4 cos x0 cos 300 +
4 sin x0 sin 300
Or
2√3 cos x0 + 2 sin x0.
The
question is can you change the form of the equation into the previous equation
be a
form of equation like 4
cos (xE-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com0tag:blogger.com,1999:blog-9077150832347230777.post-47977537269719537972019-05-01T06:53:00.003-07:002019-05-01T06:53:55.891-07:00Trigonometry Inequality
Note
the graph of the sine function f(x) = sin x0 in the interval 0 ≤ x ≤ 360 and a
line y = 1/2 below. (See Figure 1).
On the
graph, it appears that the curve with a thick line is a curve that is located
above the line y = 1/2, which is located at the interval 30 < x <150.
he
graph illustrates the values of sin x0 > 1/2.
The
discussion above is an illustration of a completion of E-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com0tag:blogger.com,1999:blog-9077150832347230777.post-19931455546637697402019-05-01T06:47:00.005-07:002019-05-01T06:47:59.034-07:00Solving Trigonometric Equations Using Basic Algebra
The discussion
of solving trigonometric equations for this time will use various methods such
as those used in algebra. This method involves factoring trigonometric forms,
converting equations into quadratic equations, or other methods often used in
solving algebraic equations. To solve trigonometric equations this form must
also be supported by your ability to apply trigonometric formulas that E-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com0tag:blogger.com,1999:blog-9077150832347230777.post-83293262447897585972019-05-01T06:45:00.001-07:002019-05-01T06:45:10.637-07:00Solve Trigonometry equations with ax + Trig bx = 0
The
discussion of this form includes forms that can be solved using the following
formulas,
sin A + sin B = 2 sin ½ (A + B) cos ½ (A – B)
sin A – sin B = 2 cos ½ (A + B) sin ½ (A – B)
cos A + cos B = 2 cos ½ (A + B) cos ½ (A – B)
cos A – cos B = -2 sin ½ (A + B) sin ½ (A – B)
Example 1.
Determine
the set of completion equations sin x0
+ sin (x – 60)0 = 0, for 0 ≤ x ≤ 360.
Answer:
sin E-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com0tag:blogger.com,1999:blog-9077150832347230777.post-44775421963625963502019-05-01T06:43:00.003-07:002019-05-01T06:43:27.839-07:00Solve Trigonometry equations with cos (x + α) - cos (x - α) = c and sin (x + α) - sin (x - α) = c
To
solve a form equation
Cos (x + α) ± cos (x – α) = c dan sin (x + α) ± sin (x –
α) = c
You
can use the sine and cosine multiplication formula that we have learned in
school, namely:
2 sin α cos β = sin (α + β) + sin (α - β)
2 cos α sin β = sin (α + β) - sin (α - β)
2 cos α cos β = cos (α + β) + cos (α + β)
- 2 sin α sin β = cos (α + β) - cos (α - β)
Example 1.
Determine the set of E-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com0tag:blogger.com,1999:blog-9077150832347230777.post-55827183190282464982019-05-01T06:37:00.002-07:002019-05-01T06:43:54.005-07:00Solve Trigonometry equations with (Trig) ax = b
Discussion of the trig form equation αx0 = b
is interpreted as a form of a trogonometric equation such as sin αx0
= b, cos αx0 = b or tan αx0 = b. In this article a more
complex form will also be developed.
Example
1.
Determine the set of completion equations
sin x0 = ½, for 0 ≤ x ≤ 360.
Answers:
Sin x0 = ½
Sin x0 = sin 300
x = 30 + n . 360, or x = (180 -30) + n . 360
for 0
≤ x E-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com0tag:blogger.com,1999:blog-9077150832347230777.post-11979414301473999482019-05-01T06:32:00.001-07:002019-05-01T06:32:30.591-07:00Definition of Trigonometric Equations
Suppose
that given the equation sin2 x + cos2 x = 1 and ask for the value of x that satisfies the
equation. It turns out that for each x value it always fulfills. Such an equation
is called identity. In trigonometry other than identity there is also a form of
equation that only applies to the values of certain variables.
In your
school you must have learned about the functions of sine, E-Pandu.Comhttp://www.blogger.com/profile/17086926200380288040noreply@blogger.com0