Thursday, December 7, 2023
HomeStudyIntroduction to antiderivatives – definition, types, formulas, and examples

# Introduction to antiderivatives – definition, types, formulas, and examples

In calculus, antiderivatives are frequently employed to determine the area under the curve. Typically, it is employed to determine the integrals of functions with respect to their integrating variables. It may be exponential, linear, logarithmic, polynomial, or constant.

In order to determine the result of a particular sort of antiderivative, antiderivatives rely heavily on limits. In this post, we will learn the concept of antiderivative as well as its types, formulas, and numerous examples.

At this level, we have learned how to calculate the derivatives of a variety of functions and have been introduced to their various applications. We now pose a question that reverses this procedure: How can we find a function f whose derivative is f, and why would we be interested in such a function, given a function f?

The first part of this question is answered by defining antiderivatives. A function with a derivative f is its antiderivative. Why are antiderivatives of interest? The requirement for antiderivatives arises in a variety of circumstances, and the remainder of the text provides numerous instances. Here, we investigate a particular instance of rectilinear motion. In our investigation of Derivatives of rectilinear motion, we demonstrated that given an object’s position function s(t), its velocity function v(t) is the derivative of s(t), or v(t) = s'(t). In addition, acceleration a(t) is the derivative of velocity v(t), or a(t) = v'(t) = s'(t).

Suppose we have access to the acceleration function a but not the velocity or position functions v and s. Since a(t)=v'(t), calculating the velocity function requires locating the acceleration function’s antiderivative. Since v(t)=s'(t), we must obtain the antiderivative of the velocity function in order to get the position function. Rectilinear motion is just one instance in which antiderivatives are required.

Many more instances will be presented throughout the remainder of the text. Let’s examine the terminology and notation for antiderivatives and determine the antiderivatives of various sorts of functions for the time being. Later in the chapter, we investigate various strategies for locating antiderivatives of more complex functions (Introduction to Techniques of Integration).

## What is the Definition of an Antiderivatives (Integral)?

An antiderivative is a function in calculus that reverses the effect of the derivative. There are numerous antiderivatives of a single function, but they are all represented in the form of a function, including the constant of integration. Integral is another name for antiderivatives. The majority of the time, however, it is related to the indefinite integral or is a crucial component of it. Antiderivatives are, in general, the inverse of derivatives. There are two primary varieties of integral (antiderivative), Namely: Definite and Indefinite.

In calculus, the definite kind of integral is commonly employed. This method of integration employs interval-like upper and lower limits for the function (r, s). The first term of the interval represents the lower limit, while the second term represents the upper limit.

### Antiderivatives are frequently encountered in physics.

• Position is the antiderivative of velocity because velocity is the derivative of position. If you know the initial location and the velocity for all time, you can calculate the position for all time.
• velocity is the antiderivative of acceleration because acceleration is the derivative of velocity. If you know the starting velocity and the acceleration for all time, you can calculate the velocity for all time.
• Newton’s law governs a significant portion of physics: force Equals mass acceleration. If you can calculate the force, you can also calculate the acceleration. By antidifferentiation, one may frequently determine the velocity and position from that point.

### This sort of integral is denoted by the formula:

rs f(z) dz = F(s) – F(r) = L

1. r & s are the function’s interval values.
2. The integrand function is f(z).
3. dz is the function’s integrating component.
4. F(s) – F(r) is the basic theorem of calculus for determining the numerical value of a function given its upper and lower limit values.
5. L is the result of applying the limit to the definite integral.

Indefinite integrals are the other sort of integral. This sort of integral lacks the function’s upper and lower limits. It is commonly used to calculate the area beneath the curve and the volume. It features the symbol “ʃ”.

ʃ f(z) dz = F(z) + c

• The integrand function is f(z).
• dz is the function’s integrating variable.
• Integrating the function yields the value F(z) for the function.
• The integral constant is C. ## How can the antiderivative be located?

By employing the rules and types of antiderivative, it is simple to solve the problems. Examples of antiderivatives are shown below. You can also use an antiderivative calculator to find the function’s antiderivative with regard to its variable.

Exemplification 1: The definite integral

Integrate [3, 5] with respect to z the expression 5z4 + 2cos(z) – 6x2z5 – z3 + 12.

Solution

Procedure 1: Apply integral notation together with the upper limit, lower limit, and integrating variable to the supplied function.

35 [5z4 + 2cos(z) – 6x2z5 – z3 + 12] dz

Procedure 2: Employ the sum, difference, and constant principles of antiderivatives to each function separately and create the definite integral notation.

35 [5z4 + 2cos(z) – 6x2z5 – z3 + 12] dz = 35 [5z4] dz + 35 [2cos(z)] dz – 35 [6x2z5] dz – 35 [z3] + 35  dz

Procedure 3: Applying the constant function rule of antiderivative and writing the constants outside integral notation is the third step.

35 [5z4 + 2cos(z) – 6x2z5 – z3 + 12] dz = 535 [z4] dz + 2 35 [cos(z)] dz – 6×235 [z5] dz – 35 [z3] + 35  dz

Procedure 4: Using the antiderivative power, constant, and trigonometric principles, integrate the preceding expression.

35 [5z4 + 2cos(z) – 6x2z5 – z3 + 12] dz = 5 [z4+1 / 4 + 1]51 + 2 [sin(z)]51 – 6×2 [z5+1 / 5 + 1]51 – [z3+1 / 3 + 1]51 + [12x]51

35 [5z4 + 2cos(z) – 6x2z5 – z3 + 12] dz = 5 [z5 / 5]53 + 2 [sin(z)]53 – 6×2 [z6 / 6]53 – [z4 / 5]53 + [12z]53

35 [5z4 + 2cos(z) – 6x2z5 – z3 + 12] dz = 5/5 [z5]53 + 2 [sin(z)]53 – 6×2/6 [z6]53 – 1/5 [z4]53 + [12z]53

35 [5z4 + 2cos(z) – 6x2z5 – z3 + 12] dz = [z5]53 + 2 [sin(z)]53 – x2 [z6]53 – 1/5 [z4]53 + 12[z]53

Procedure 5:The result is obtained by applying the fundamental theorem rs f(z) dz = F(s) – F(r).

35 [5z4 + 2cos(z) – 6x2z5 – z3 + 12] dz = [55 – 35] + 2 [sin (5) – sin (3)] – x2 [56 – 36] – 1/5 [54 – 34] + 12[5 – 3]

35 [5z4 + 2cos(z) – 6x2z5 – z3 + 12] dz = [3125 – 243] + 2 [sin (5) – sin (3)] – x2 [15625 – 729] – 1/5 [625 – 81] + 12[5 – 3]

35 [5z4 + 2cos(z) – 6x2z5 – z3 + 12] dz =  + 2 [sin (5) – sin (3)] – x2  – 1/5  + 12

35 [5z4 + 2cos(z) – 6x2z5 – z3 + 12] dz = 2882 + 2sin (5) – 2sin (3) – 14896×2 – 544/5 + 24 35 [5z4 + 2cos(z) – 6x2z5 – z3 + 12] dz = 2882 + 2sin (5) – 2sin (3) – 14896×2 – 108.8 + 24

35 [5z4 + 2cos(z) – 6x2z5 – z3 + 12] dz = 2773.2 + 2sin (5) – 2sin (3) – 14896×2 + 24

35 [5z4 + 2cos(z) – 6x2z5 – z3 + 12] dz = 2797.2 + 2sin (5) – 2sin (3) – 14896×2

Illustration 2: For indefinite integral

Integrate 4z3 – 12sin(z) – 15z4 + 17z5 + 12z with respect to z.

Solution

Procedure 1: Apply integral notation together with the upper limit, lower limit, and integrating variable to the supplied function.

ʃ [4z3 – 12sin(z) – 15z4 + 17z5 + 12z] dz

Procedure 2: Apply the sum, difference, and constant principles of antiderivatives to each function separately and create the definite integral notation.

ʃ [4z3 – 12sin(z) – 15z4 + 17z5 + 12z] dz = ʃ [4z3] dz – ʃ [12sin(z)] dz – ʃ [15z4] dz + ʃ [17z5] dz + ʃ [12z] dz

Procedure 3: the constant function rule of antiderivative, write the constants outside integral notation in the third step.

ʃ [4z3 – 12sin(z) – 15z4 + 17z5 + 12z] dz = 4 ʃ [z3] dz – 12 ʃ [sin(z)] dz – 15 ʃ [z4] dz + 17 ʃ [z5] dz + 12 ʃ [z] dz

Procedure 4: Using the power and trigonometric laws of antiderivative, integrate the preceding expression.

ʃ [4z3 – 12sin(z) – 15z4 + 17z5 + 12z] dz = 4 [z3+1 / 3 + 1] – 12 [-cos(z)] – 15 [z4+1 / 4 + 1] + 17 [z5+1 / 5 + 1] + 12 [z1+1 / 1 + 1] + C

ʃ [4z3 – 12sin(z) – 15z4 + 17z5 + 12z] dz = 4 [z4 / 4] – 12 [-cos(z)] – 15 [z5 / 5] + 17 [z6 / 6] + 12 [z2 / 2] + C

ʃ [4z3 – 12sin(z) – 15z4 + 17z5 + 12z] dz = 4/4 [z4] – 12 [-cos(z)] – 15/5 [z5] + 17/6 [z6] + 12/2 [z2] + C

ʃ [4z3 – 12sin(z) – 15z4 + 17z5 + 12z] dz = z4 + 12cos(z) – 3z5 + 17z6/6 + 6z2 + C

## Frequently Asked Questions on Integration

### What is the meaning of integration?

The process of locating the antiderivative of a function is integration. In a similar fashion, the slices are assembled to form the whole. The process of integration is the opposite of differentiation. ### What purpose does integration serve?

Integration is utilized to determine the volume, area, and central values of several things.

### What are integration’s practical applications?

Integrations are essential for calculating the center of gravity, center of mass, and predicting the positions of planets, etc.

### What is calculus’ fundamental theorem?

The fundamental theorem of calculus connects the concepts of function differentiation and integration.

### Mention two distinct integral types in mathematics.

Integration is one of the two most important mathematical concepts, and the integral assigns a number to the function. The two distinct integral kinds are the definite integral and the indefinite integral.

## Conclusion

This article covered the whole fundamentals of antiderivatives. By reading the preceding article, you may now grasp the fundamental ideas of integrals. Using the formulas and examples provided in this page, you may easily solve problems with definite or indefinite integrals.

RELATED ARTICLES

### 10 Best Nursing Schools in Ontario

SCHOLARSHIP UPDATE

Be the First to hear about new Scholarships. Set a Reminder now. Never miss an Opportunity.