Integral Calculus: An introduction to its definition, types, and examples

In mathematics, integral calculus is a type of calculus that is used to integrate the functions to find the area under the curve. Calculus is a wide concept used in mathematics, engineering, and other fields of science. 

This subtype of calculus is a rather difficult topic for students. They face many difficulties while solving the complex problems of calculus. In this post, we will describe the definition, types, formulas, and examples of integral calculus with examples and solutions.  

What is integral calculus?

Integral calculus is a branch of calculus that deals with the applications and theory of integrals. It is also known as the antiderivative and is defined as a function whose derivative (rate of change) is equal to the function that is being integrated. 

It is also defined as the term that is used to calculate the new function whose original function is derivative or to find the numerical value of the function with the help of boundary values. 

Types of integral calculus

There are two main types or methods of integral calculus. Here are those two types:

• Definite integral
• Indefinite integral

The definite integral is a type of integration that is helpful in determining the numerical value of the function by taking the upper and lower limits of the function. The fundamental theorem is used to apply the upper & lower limit values to the integrated function. 

The indefinite integral is that type of integration that is used to calculate the new function whose original function is differential (rate of change). In this type of integral, the upper and lower limit values are not involved. 

Formulas of integral calculus

The general expression for calculating the definite integral is:

The general expression for calculating the indefinite integral is:

How to calculate the problems of integral calculus?

The problems of integral calculus can be solved either by using an integration calculator or a manual method. Let us learn how to calculate the problems manually and using an online calculator. 

• By using an integral calculator 

The integral calculator is an online resource that is helpful in calculating the problems of integral calculus with steps in no time. You have to put the data values of the function and the result will come in a fraction of a second after clicking the calculate button.  

• Solving integral problems manually

Here are a few examples of calculating the problems of integral calculus that are solved by a manual method. 

Example 1

Evaluate the indefinite integral by integrating the given function with respect to “v”.

f(v) = 3v⁵ + 12v⁶ – 14v⁴ + 5cos(v) + 11r – 6 

Solution 

Step 1: Take the given function, integrating variable, and apply the notation of integral to it.

f(v) = 3v⁵ + 12v⁶ – 14v⁴ + 5cos(v) + 11r – 6 

integrating variable = v

ʃ f(v) dv = ʃ [3v⁵ + 12v⁶ – 14v⁴ + 5cos(v) + 11r – 6] dv 

Step 2: Now use the sum and difference rules of integral calculus to apply the notation to each term of the function separately to make the calculations easier. 

ʃ [3v⁵ + 12v⁶ – 14v⁴ + 5cos(v) + 11r – 6] dv = ʃ [3v⁵] dv + ʃ [12v⁶] dv – ʃ [14v⁴] dv + ʃ [5cos(v)] dv + ʃ [11r] dv – ʃ [6] dv 

Step 3: Now use the constant function rule of integral calculus to take constant coefficients outside the integral notation. 

ʃ [3v⁵ + 12v⁶ – 14v⁴ + 5cos(v) + 11r – 6] dv = 3ʃ [v⁵] dv + 12ʃ [v⁶] dv – 14ʃ [v⁴] dv + 5ʃ [cos(v)] dv + 11ʃ [r] dv – ʃ [6] dv 

Step 4: Use the power and trigonometry rules of integral calculus to integrate the above expression.  

ʃ [3v⁵ + 12v⁶ – 14v⁴ + 5cos(v) + 11r – 6] dv = 3 [v⁵⁺¹ / 5 + 1] + 12 [v⁶⁺¹ / 6 + 1] – 14 [v⁴⁺¹ / 4 + 1] + 5 [sin(v)] + 11 [r¹⁺¹ / 1 + 1] – [6v] + C

ʃ [3v⁵ + 12v⁶ – 14v⁴ + 5cos(v) + 11r – 6] dv = 3 [v⁶ / 6] + 12 [v⁷ / 7] – 14 [v⁵ / 5] + 5 [sin(v)] + 11 [r² / 2] – [6v] + C

ʃ [3v⁵ + 12v⁶ – 14v⁴ + 5cos(v) + 11r – 6] dv = 3/6 [v⁶] + 12/7 [v⁷] – 14/5 [v⁵] + 5 [sin(v)] + 11/2 [r²] – [6v] + C

ʃ [3v⁵ + 12v⁶ – 14v⁴ + 5cos(v) + 11r – 6] dv = 0.5 [v⁶] + 1.71 [v⁷] – 2.8 [v⁵] + 5 [sin(v)] + 5.5 [r²] – [6v] + C

ʃ [3v⁵ + 12v⁶ – 14v⁴ + 5cos(v) + 11r – 6] dv = 0.5v⁶ + 1.71v⁷ – 2.8v⁵ + 5sin(v) + 5.5r² – 6v + C

Example 2: 

Calculate the definite integral by integrating the given function with respect to “t”.

f(t) = 12t⁵ – 2t³ + 9t² + 12t + 12 in the interval of [1, 2].

Solution 

Step 1: Take the given function, integrating variable, and apply the notation of integral to it.

f(t) = 12t⁵ – 2t³ + 9t² + 12t + 12

integrating variable = t

Step 2: Now use the sum and difference rules of integral calculus to apply the notation to each term of the function separately to make the calculations easier.

Step 3: Now use the constant function rule of integral calculus to take constant coefficients outside the integral notation.

Step 4: Use the power rule of the integral calculus to integrate the above expression.

Step 5: Use the fundamental theorem of calculus to apply the upper and lower limit.

Relevant Links
Interview IITian’s journey from the JEEs till graduation -Examgyani
Is Coaching Important to Crack a Competitive Exam
Is Engineering in India a Right Career Option in 2023?

Wrap Up

Now you can grab almost all the basic intent of integral calculus as we have discussed the definition, types, formulas, and solved examples of integral calculus. Now you can solve the problems of integral calculus either by using an online calculator or manually.