Integral in calculus: Definition, rules, and examples

In mathematics, the integral is used frequently in calculus to integrate complex functions. It determines the graph of the given functions for various problems. It can integrate the constant, linear, or polynomial function with respect to corresponding integrating variables.

 Limit calculus is also used to define the integral. In this article, we’ll go through the basic definition and rules of integration with the help of examples.

What is integral?

According to Britannica:

In mathematics, an integral is used to find the definite or indefinite integral of the given function using the interval of the function. The numerical value of the integral for some interval is the area under the graph of a function known as a definite integral.

While the function is indefinite integral whose derivative makes the original function. In simple words, integrating the given function for some interval (upper or lower limits) is known as a definite integral while integrating the function without applying the limits is known as an indefinite integral.

The main work of the integral calculus is to define the volume, area, displacement, and some other concepts in calculus. The process of finding the definite or indefinite integrals of the given functions is known as integration.

Equations of integral

Integrals use different equations for the definite and indefinite integral with the help of integral notation. The notation ʃ is used in integrals. The upper and lower limits are used in the definite integral.

The equation for the definite integral is given below

uv f(y) dy 

  • In the notation of definite integral uv  u is the lower limit & v is the upper limit of the given function.
  • f(y) is the function to be integrated.
  • dy is the variable of integration.
  • M is the numerical value of the given function after substituting the upper and lower limits by using the fundamental rule of calculus.

The equation of the indefinite integral is:

ʃ f(y) dy = F(y) + c

  •  f(y) is the function to be integrated or simply integrand.
  • dy is the variable of integration.
  • F(y) is the result of the given function after integrating. 
  • C is a constant of integral

You can use an antiderivative calculator to integrate functions by using the definite or indefinite method. You’ll get the step-by-step solution of the given function by using this tool.

e

Rules of integration 

There are various rules of integration used to integrate the functions. Let us discuss the rules of integration with examples.

Constant rule

According to the constant rule of the integration, the result will be the corresponding variable of the function multiplied by that constant. The constant function doesn’t become zero in integration. The equation of the constant function is:

ʃ C dx = Cx + C1

Following are a few examples solved by using the sum rule.

Example 1: Foe indefinite integral

Integrate 36 with respect to x.

Solution 

Step 1: Write the given values.

f(x) = 36

integrating variable = dx

Step 2: Now take the general equation of the constant rule.

ʃ C dx = Cx + C1

step 3: Put the above values in the equation of the constant rule.

ʃ 36 dx = 36x + C1

Example 2: For definite integral

Integrate 48 with respect to x having limits from 4 to 5.

Solution 

Step 1: Write the given values.

f(x) = 48

lower limit = u = 4

upper limit = v = 5

integrating variable = dx

Step 2: Now take the general equation of the constant rule.

uv C dx 

step 3: Put the above values in the equation of the constant rule.

45 48 dx 

Step 4: Now integrate the above expression and apply the fundamental theorem of calculus.

45 48 dx = 48[x]54

45 48 dx = 48[5 – 4]

45 48 dx = 48[1]

45 48 dx = 48

Power rule

According to the power rule of integration, the corresponding variable’s power increased by one, and write the power of the variable in the denominator of the variable. The equation of the power rule is:

ʃ [f(x)]n dx = [f(x)]n+1/n + 1 + C

Example 

integrate 2x3 with respect to x.

Solution 

Step 1: Write the given values.

f(x) = 2x3

integrating variable = dx

Step 2: Now take the general equation of the constant rule.

ʃ [f(x)]n dx = [f(x)]n+1/n + 1 + C

step 3: Put the above values in the equation of the power rule to integrate the given function.

ʃ 2x3 dx = 2x3+1/3 + 1 + C

ʃ 2x3 dx = 2x4/4 + C

ʃ 2x3 dx = x4/2 + C

Sum rule

According to the sum rule of integration, the notation of the integral must be applied to all the functions separately. This rule holds for both types of integral. After applying the limit notation separately integrate the functions to get the result.

The equation for the sum rule of integral is:

ʃ (f(x) + g(x)) dx = ʃ f(x) dx + ʃ g(x) dx

Following are a few examples solved by using the sum rule.

Example 1: For indefinite integral

Integrate sin(x) + x2 with respect to x, 

Solution 

Step 1: Write the given values.

f(x) = sin(x)

g(x) = x2

integrating variable = dx

Step 2: Take the equation of the sum rule.

ʃ (f(x) + g(x)) dx = ʃ f(x) dx + ʃ g(x) dx

Step 3: Now put the given values in the equation of the sum rule.

ʃ (sin(x) + x2) dx = ʃ sin(x) dx + ʃ x2 dx

Step 4: Integrate the above expression by using the trigonometric and power rule.

ʃ (sin(x) + x2) dx = ʃ sin(x) dx + ʃ x2 dx

ʃ (sin(x) + x2) dx = -cos(x) + x2+1 / 2 + 1 + C

ʃ (sin(x) + x2) dx = -cos(x) + x3 / 3 + C

Example 2: For definite integral

Integrate 3x + x3 with respect to x from 2 to 3. 

Solution 

Step 1: Write the given values.

f(x) = 3x

g(x) = x3

upper limit = v = 3

lower limit = u = 2

integrating variable = dx

Step 2: Take the equation of the sum rule.

uv (f(x) + g(x)) dx = uv f(x) dx +uv g(x) dx

Step 3: Now put the given values in the equation of the sum rule.

23 (3x + x3) dx = 23 3x dx + 23 x3 dx

Step 4: Integrate the above expression by using the power rule.

23 (3x + x3) dx = 23 3x dx + 23 x3 dx

23 (3x + x3) dx = [3x1+1 / 1 + 1]32 + [x3+1 / 3 + 1]32 

23 (3x + x3) dx = [3x2 / 2]32 + [x4 / 4]32 

23 (3x + x3) dx = 3/2[x2]32 + 1/4[x4]32 

Step 5: Now apply the limit values by using the fundamental theorem of calculus.

23 (3x + x3) dx = 3/2[32 – 22] + 1/4[34 – 24

23 (3x + x3) dx = 3/2[9 – 4] + 1/4[81 – 16] 

23 (3x + x3) dx = 3/2[5] + 1/4[65] 

23 (3x + x3) dx = 15/2 + 65/4 

23 (3x + x3) dx = 7.5 + 16.25

23 (3x + x3) dx = 23.75 

The difference rule of integration works similar to sum rule but having the minus sign. For example, ʃ (f(x) – g(x)) dx = ʃ f(x) dx – ʃ g(x) dx

Summary 

In this post, we have learned about the basics of integration. Now you are witnessed that integral is not a difficult topic. You can easily solve the problems of integral just by learning the above-mentioned rules and equations of integral.

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Updated: April 24, 2022 — 4:35 am

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