What are Definite and Indefinite Integrals and how to calculate them?

In calculus, definite and indefinite integrals are the types of integral. Integral is usually defined as the reverse process of differentiation. In simple words, integration inverts that the differentiation does. Integration is a process that is used to find the area under the curve.

There are wide uses of integration in mathematics, physics, and the other branches of science. Integration can be applied to calculate the area, central points, and volumes of the curves or curved surfaces.

In this post, we will learn about the types of integration, their working, and how to calculate them.

What is the Definite Integral?

Definite integral, in calculus, is the type of the integral. In definite integral, there are upper limits and lower limits to be applied to the problem. The main purpose of this type of integral is to find the difference between the integral values of the function that is provided.

The definite integral is that type, in which first we have to apply integral and get our required results and then put upper and lower limits by using a fundamental theorem of calculus. By applying this theorem, we can calculate the limits of the integration.

According to the fundamental theorem of calculus.

f(x) dx = [F(x)]vu = F(v) – F(u)

How to find Definite Integral?

For doing this type of calculation, we have to know all the basic rules to calculate integrals. First of all, calculate the integral of the given problem by using different rules of integral. And then apply limits according to the fundamental theorem of calculus.

For such calculations, there are several tools that lessen the difficulty of performing large calculations. One of the tool is listed below.

https://www.meracalculator.com/math/integral.php

Example 1

Calculate the integral by using the definite integral of (2x5 + 5x) dx?

Solution

Step 1: Write the given integral function.

(2x5 + 5x) dx

Step 2: Apply the sum of the function in the integral notation rule.

(2x5 + 5x) dx = 2x5 dx + 5x dx

Step 3: Take the constant outside of the integral notation.

(2x5 + 5x) dx = 2x5 dx + 5x dx

Step 4: Apply the power rule of integration on the above equation.

(2x5 + 5x) dx = 2(x5+1/5 + 1)61 – 5(x1+1/1 + 1)61

= 2  – 5

= 2/6  – 5/2

= 1/3  – 5/2

Step 5: Apply the upper and lower limits by using the fundamental theorem of calculus.

f(x) dx = [F(x)]vu = F(v) – F(u)

(2x5 + 5x) dx = 1/3 (66 – 16) – 5/2 (62 – 12)

(6x4 + 3x) dx = 1/3 (46656 – 1) – 5/2 (36 – 1)

= 1/3 (46655) – 5/2 (35)

= 15551.667 – 5(17.5)

= 15551.667 – 87.5

(6x4 + 3x) dx= 15464.167

Example 2

Calculate the integral by using the definite integral of (3x5 + 5x4 – 2) dx?

Solution

Step 1: Write the given integral function.

(3x5 + 5x4 – 2) dx

Step 2: Apply the difference of the function in the integral notation rule.

(3x5 + 5x4 – 2) dx = 3x5 dx + 5x4 dx – 2 dx

Step 3: Take the constant outside of the integral notation.

(3x5 + 5x4 – 2) dx = 3x5 dx + 5x4 dx – 2 dx

Step 4: Apply the power rule of integration on the above equation.

(3x5 + 5x4 – 2) dx = 3(x5+1/5 + 1)31 + 5(x4+1/4 + 1)31 – 2(x)31

= 3  + 5  – 2(x)31

= 3/6  + 5/5  – 2(x)31

= 1/2  +  – 2(x)31

Step 5: Apply the upper and lower limits by using the fundamental theorem of calculus.

f(x) dx = [F(x)]vu = F(v) – F(u)

(3x5 + 5x4 – 2) dx = 1/2 (36 – 16) + (35 – 15) – 2(3 – 1)

(3x5 + 5x4 – 2) dx = 1/2 (729 – 1) + (243 – 1) – 2(2)

= 1/2 (728) + (242) – 4

= 728/2 + 242 – 4

= 364 + 238

= 602

What is Indefinite Integral?

Indefinite integral, in calculus is another type of the integral. In indefinite integral, we have to calculate integrals without applying limits. It is mainly used to invert the differentiation functions. Indefinite integral is also written by the name of antiderivative.

The indefinite integral is that type of integration in which we have to apply all the rules and get the required result of the function. In indefinite integral, there are no upper and lower limits. We have to calculate only the function to determine the integral.

ʃ f(x) dx = F(x) + C

In this equation, f(x) is the function, F(x) is integral, and c is the arbitrary constant.

How to find Indefinite Integral?

Like definite integral, we can find the indefinite integral by applying various rules of the integral. The only difference among them is that in definite integral, we apply limits while in indefinite integral, we do not have to apply limits.

Example 1

Find the antiderivative of given indefinite integral 3x8 – 5y4 + 19z?

Solution

Step 1: Write the given integral function.

3x8 – 5y4 + 19z

Step 2: Now write the given problem in integral notation.

ʃ (3x8 – 5y4 + 19z) dx

Step 3: Apply sum and difference rule of the integral on the given function.

ʃ (3x8 – 5y4 + 19z) dx = ʃ 3x8 dx – ʃ 5y4 dx + ʃ 19z dx

Step 4: Take constant outside of the integral notation.

ʃ (3x8 – 5y4 + 19z) dx = 3ʃ x8 dx – 5y4 ʃ dx + 19z ʃ dx

Step 5: Apply the power rule of integration on the above equation.

ʃ (3x8 – 5y4 + 19z) dx = 3 (x8+1/8 + 1) – 5y4 (x) + 19z (x) + C

ʃ (3x8 – 5y4 + 19z) dx = 3 (x9/9) – 5xy4 + 19xz + C

ʃ (3x8 – 5y4 + 19z) dx = 3/9 x9 – 5xy4 + 19xz + C

ʃ (3x8 – 5y4 + 19z) dx = 1/3 x9 – 5xy4 + 19xz + C

To avoid such large and time-consuming calculations, you can use antiderivative calculator with steps for the calculation of the given problem related to the integral.

Summary

Both types of integrals are very important in calculus. For solving the problems, to calculate the area under the curve or the volumes, these types are very essential to perform integral calculations.

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