One of the most important branches of mathematics includes calculus. Calculus is a method of systematically calculating problems, which deals with finding properties or values of functions by integrals and derivatives.

## Key Takeaways

- Definite integrals calculate the signed area under a curve within a specific interval, providing a numerical value.
- Indefinite integrals determine the antiderivative of a function, expressing the result as a family of functions with an added constant.
- Both definite and indefinite integrals are important concepts in calculus, but they serve different purposes: definite integrals quantify areas, while indefinite integrals explore antiderivatives.

**Definite vs. Indefinite Integrals**

The difference between Definite and Indefinite Integral is that a definite integral is defined as an integral that has upper and lower limits and has a constant value as the solution; on the other hand, an indefinite integral is defined as an internal which do not have limits applied to it, and it gives a general solution for a problem.

A definite integral of a function of an unknown variable is the representation of a number with upper and lower limits. An indefinite integral is the representation of a family of functions without limits.

**Comparison Table**

Parameter of Comparison | Definite Integrals | Indefinite Integrals |
---|---|---|

What it means | A definite integral has lower and upper limits and, on solving, gives a constant result. | An indefinite integral is an integral with no limits, and a mandatory arbitrary constant is added to the integral. |

What it represents | The definite integral represents the number when its upper and lower limits are constant. | An indefinite integral generally represents a family of various functions with derivatives f. |

Limits applied | The upper and lower limits applied in a definite integral are always constant. | In indefinite integral, there are no limits since it is a general representation. |

Solution obtained | The values or solutions obtained from definite integrals are constant. However, they can either be positive or negative. | The solution of an indefinite integral is a general solution with a constant value added to it, which is generally represented by C. |

Used for | A definite integral is widely used in physics and engineering. Some of the areas of use of a definite integral include calculating values of force, mass, work, areas between curves, volumes, act length of curves, surface areas, moments and center of mass, exponential growth, decay, etc. | Indefinite integrals are used in fields like business and sciences, including engineering, economics, etc. It is used where a general solution is required for a problem. |

**What is a Definite Integral?**

A definite integral represents a number that gives a constant result. A definite integral always has an upper limit and a lower limit.

The solution can either be positive or negative. The solution obtained from a definite integral always lies in a specific area.ย

Some areas where definite integrals are used are a calculation of work, force, mass, areas, surface areas, the area between curves, length of arcs, moments, the center of mass, exponential growth and decay, etc.

**What is Indefinite Integral?**

An indefinite integral is defined as the integral without limits. The indefinite integral represents a family of various functions having derivative f.ย

The solution obtained by solving the unknown function of an indefinite integral is a generalized solution; therefore, it also has variables. The area of the solution of an indefinite integral is not specified.ย

Indefinite integrals are used where a general solution to the problem is required. Indefinite integrals are used in business, sciences, engineering, economics, etc.

**Main Differences Between Definite and Indefinite Integral**

- A definite integral can be defined as an integral with limits; conversely, an indefinite integral can be defined as an integral without limits.
- A definite integral represents the number with constant upper and lower limits. In contrast, an indefinite integral represents the general solution for a family of functions having derivative f.

**References**

- https://www.tandfonline.com/doi/abs/10.1080/10652469.2014.1001385
- https://www.koreascience.or.kr/article/JAKO200931559904911.page

Emma Smith holds an MA degree in English from Irvine Valley College. She has been a Journalist since 2002, writing articles on the English language, Sports, and Law. Read more about me on her bio page.