Indefinite Integral: Definition, Formulas, Properties and Examples of Problems
Indefinite Integral: Definition, Formulas, Properties and Examples of Problems – What is meant by Indefinite Integral and how to calculate the mathematical operation? At this time About the knowledge.co.id will discuss what is an Indefinite Integral and the things that surround it. Let's look at the discussion in the article below to understand it better.
Indefinite Integral: Definition, Formulas, Properties and Examples of Problems
Integral is a form of mathematical operation which is the reverse or also known as the inverse of the derivative operation. As well as the limit of the amount or a certain area.
There are two kinds of things that must be carried out in an integral operation, both of which have been categorized into 2 types of integrals. Among other things: the integral as an inverse or the opposite of a derivative or what is commonly referred to as an Indeterminate Integral. As well as the second, the integral as the limit of the number or area of a certain area which is referred to as a definite integral.
Indefinite integral (English: indefinite integral) or antiderivative is a form of integration operation of a function that produces a new function. This function does not yet have a definite value (in the form of a variable) so that the method of integration that produces this indefinite function is called "indefinite integral".
If f is an indefinite integral of a function F then F'= f. The process of solving antiderivatives is antidifferentiation. Antiderivatives are definitely related integral through the “Fundamental theorem of calculus”, and provides an easy way to calculate integrals of various function.
As previously mentioned, Indefinite Integral or what is commonly referred to as Indefinite Integral or there is also those who call it an Antiderivative is a form of integration operation on a function that produces a function new.
This function does not have a definite value until the method of integration which produces this indefinite function is called an indefinite integral. If f is an indefinite integral of a function F then F'= f.
The process of solving the antiderivative is the antidifferentiation of the antiderivative which is related to the integral by the "Fundamental Theorem of Calculus". As well as providing an easy way to calculate the integral of various functions.
As previously explained, the indefinite integral in mathematics is the inverse of the derivative. The derivative of a function, when integrated, will produce the function itself.
Let's take a good look at some examples of derivatives in algebraic functions below:
- The derivative of the algebraic function y = x3 is yI = 3x2
- The derivative of the algebraic function y = x3 + 8 is yI = 3x2
- The derivative of the algebraic function y = x3 + 17 is yI = 3x2
- The derivative of the algebraic function y = x3 – 6 is yI = 3x2
As we have learned in the derivative material, variables in a function will experience demotion.
Based on the example above, we can see if there are many functions that have the same derivative, namely yI = 3x2.
The function of the variable x3 as well as the function of the variable x3 that are subtracted or added to a number (for example: +8, +17, or -6) have the same derivative.
If we integrate the derivatives, then they should be the initial functions before they are derived.
However, in cases where the initial function of a derivative is not known, then the integral result of the derivative can be written as:
f(x) = y = x3 +C
With a value of C can be anything. C notation is also referred to as integral constant. The indefinite integral of a function is denoted as follows:
In the notation above we can read the integral to x". notation is called the integral. In general, the integral of the function f (x) is the sum of F(x) with C or:
Because integrals and derivatives are related to each other, the integral formula can be obtained from the reduction formula. If derivative:
Then the algebraic integral formula is obtained:
provided that n ≠ 1
As an example consider some of the following algebraic integral functions:
- How to Read an Indefinite Integral
After reading the description above, do you know how to read integral sentences? The integral reads like this:
read Indefinite Integral of Function f (x) to Variable X.
Integral General Formula
The following are the general formulas for integrals:
- Integral Formula Development
Let's take a good look at some examples of derivatives in algebraic functions below:
- The derivative of the algebraic function y = x3 is yI = 3x2
- The derivative of the algebraic function y = x3 + 8 is yI = 3x2
- The derivative of the algebraic function y = x3 + 17 is yI = 3x2
- The derivative of the algebraic function y = x3 – 6 is yI = 3x2
Integral Properties
The properties of the integral include:
- ∫ k. f(x)dx = k. ∫ f (x) dx (where k is a constant)
- ∫ f (x) + g (x) dx = ∫ (x) dx + ∫ g (x) dx
- ∫ f (x) – g (x) dx = ∫ f (x) dx – ∫ g (x) dx
Determine the Curve Equation
The gradient as well as the equation of the tangent to the curve at a point.
If y = f (x), the slope of the tangent to the curve at any point on the curve is y’ = = f'(x).
Therefore, if the gradient of the tangent line is known, the curve equation can be determined in the following way:
y = ∫ f ‘ (x) dx = f (x) + c
If one of the points passing through the curve is known, the value of c can also be known so that the equation of the curve can be determined.
Integral Problem Example
Problem 1
Discussion
In this problem, the upper bound is 1 and the lower bound is -2. The first step we need to do is perform the integral of the 3x function2 + 5x + 2 to be like below.
Once we get the integral form of the function, we can plug in the upper and lower bound values into the function and then reduce them as follows.
The result of the integral is 27.5.
Problem 2.
It is known that the derivative y = f (x) is = f '(x) = 2x + 3
If the curve y = f (x) passes through the point (1, 6), then determine the equation of the curve.
Answer:
f'(x) = 2x + 3.
y = f (x) = ʃ (2x + 3) dx = x2 + 3x + c.
The curve passes through point (1, 6), meaning f (1) = 6 so that the value of c can be determined, namely 1 + 3 + c = 6 ↔ c = 2.
So, the equation of the curve in question is:
y = f(x) = x2 + 3x + 2.
Problem 3.
Look for the result of ʃ21 6x2 dx !
Discussion
So, the result of ʃ21 6x2 dx is 14.
Problem 4
The slope of the tangent to the curve at the point (x, y) is 2x – 7. If the curve passes through the point (4, –2), then determine the equation of the curve.
Answer:
f'(x) = = 2x – 7
y = f (x) = ʃ (2x – 7) dx = x2 – 7x + c.
Because the curve passes through the point (4, –2)
so:
f (4) = –2 ↔ 42 – 7(4) + c = –2
–12 + c = –2
c = 10
So, the equation of the curve is:
y = x2 – 7x + 10.
What is the value of the definite integral of ʃ-2-2 3x2 – 2x + 1dx ?
Discussion
So, the definite integral value of ʃ-2-2 3x2 – 2x + 1 dx is 20.
Problem 5.
Calculate the definite integral of ʃ94 1/√x dx !
Discussion
So, the definite integral value of ʃ94 1/√x dx is 2.
Thus the review from About the knowledge.co.id about Indefinite integral, hopefully can add to your insight and knowledge. Thank you for visiting and don't forget to read other articles
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