One of the most important branches of mathematics includes calculus. Calculus is a manner of calculation of problems in a systematic way which usually 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 the integral which has upper and lower limits and has a constant value as the solution, on the other hand, an indefinite integral is defined as the internal which do not have limits applied to it and it gives a general solution for a problem.

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A definite integral of a function of an unknown variable is the representation of a number which has 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 is the one that has lower and upper limits and on solving gives a constant result. | An indefinite integral is the integral in which no limits are applied and has a mandatory arbitrary constant added to the integral. |

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

Limits applied | The upper and the 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 and it has 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 calculation of values of force, mass, work, areas between curves, volumes, act length of curves, surface areas, moments and center of mass, exponential growth and decay, etc. | Indefinite integrals are of use in fields like business, sciences including engineering, economics, etc. It is used in areas where a general solution is required for a problem. |

**What is a Definite Integral?**

A definite integral is defined as the representation of a number which 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.

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, 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 is the representation of a family of various functions having derivative f.

The solution obtained on solving the unknown function of an indefinite integral is a generalized solution and therefore it also has variables in it. The area of the solution of an indefinite integral is not specified.

Indefinite integrals are used at places 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 the integral which has limits, on the other hand, an indefinite integral can be defined as the integral without limits.
- A definite integral is the representation of the number when it has upper and lower limits constant whereas an indefinite integral is the representation of 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.