Table of contents

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Introduction:

Integration is one of the main concepts within calculus. The reason is that these concepts are used in various different fields. Primarily, integration has 2 types which are:

  1. Indefinite Integral
  2. Definite Integral

This article we’ve discussed what indefinite integrals and definite integrals are, how indefinite integrals and definite integrals are represented. You’ll also be able to find the boundary values and the way through which you can calculate definite integral and indefinite integral.

Indefinite Integral:

Indefinite function is a function which produces a function or family of function rather than any real number value after evaluating it.

It means to take the antiderivative of the function with respect to some dependent variable depending on the function.

Indefinite integral doesn’t have defined or predefined limits instead it is always expressed as without the limits.

Indefinite integrals contain an arbitrary constant or we have to add them when we are dealing with indefinite integrals. Reason of adding arbitrary values is that any specific value will therefore make a valid antiderivative.

Also find other integration tools on this website which can help you a lot in learning more about the concept. We offer volume of revolution calculator and cylindrical shell volume calculator along with cross sectional area calculator with steps.

Representation of Indefinite Integral:

The indefinite integral represents the integration symbol, function and dx. The notation of writing or representing definite integral is given as follow:

$$ \int f(x)dx \Rightarrow F(x) + c $$

By looking at the above example, we can see that f(x) is the antiderivative under the sign of integration which gives us a function after evaluation.

Related: Online tools are modern way of learning complex calculations by doing run time practice. You can use multiple integral calculator and triple integral solver on this website and learn and practice easily online.

Boundary Values:

In the above representation of indefinite integral we noted that there isn’t any boundary value of integral when it comes to indefinite integral because it does not have any limit.

As there is no limit of indefinite integral, therefore it always gives us a general function or family of functions for evaluating it.

Related: Learn what you need to know about partial fraction from this blog post.

How to Solve:

For evaluating indefinite integral, we have to simply take the anti-derivative of integrand under the integration for finding the function. The below example indefinite integral helps us to understand it properly:

Example:

$$ \int_2^3 x^3 = | \frac{x^4}{4} | $$

In this example, you can see we have calculated a function after the integration with a constant which is called an arbitrary constant of integration.

Definite Integral:

As the name of definite integral shows, definite integral will gives you a definite number or some real value. Unlike indefinite integral, the definite integral has limits.

So, the definite integral of a function is a definite number or one single number.

It's just like finding the area under the curve, whenever you find an area it will be some definite number which shows the quantity.

But the conditions here are that f(x) function under the integral should always be positive. But if the function is not positive, we will have to do a signed area under the curve which highlights the positive area on the top and negative area on the bottom.

Moreover, The definite integral has a boundary value defined for finding the area under the curve.

Representation of definite Integral:

The definite integral looks the same as the indefinite integral where we can see the integration symbol, function and dx. But you can see additional values on top and bottom of the integration symbol. These values are the limits. The notation of writing or representing definite integral are given as follow:

$$ \int_a^b f(x) dx $$

There is not any complex difference between writing definite and indefinite integral.

Simply, if you have given upper and lower limits to integrals like a&b in our case, then the integral must be a definite integral.

After evaluating that integral of the function f(x), the result will always be a single definite number under the given boundaries like a&b in our case.

I.e

$$ \int_a^b f(x) dx \Rightarrow \;\;\; \text{Number} $$

Boundary Values:

As we see, definite integral has a boundary value written at its integral, which is the main thing that differs it from indefinite integral. These boundary values are called the upper limit and lower limit of that integral.

Upper limit: The number at the top of the integral is called upper limit i.e b.

Lower limit: The number at the bottom of the integral is called lower limit i.e a.

To learn specifically about the limits, find limits calculator with steps free. Also you can use integral solver on home page to evaluate integration problems.

How To Solve:

Well for solving the definite integral there are two simple steps that we have to follow:

Let’s see an example for proper understanding of how to evaluate indefinite integrals.

Example:

$$ \int_2^3 x^3 = | \frac{x^4}{4} | $$

The above example helps us to understand the evaluation of indefinite integral. We hope this article helped you in order to learn what is a definite integral and what is indefinite integral.

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