What is an improper integral calculator ?

So as we tread on into high school with math, calculus becomes our roommate. Wherever you go, whatever you do, you find calculus- be it math, physics, chemistry, economics, statistics, or anything in general. So, when we come to integration, we call the term integrate, integrals. Moreover, there can be roughly two kinds of integrals- definite and indefinite. Hence, for an indefinite integral, you do not have any lower or upper limit concerning which you need to calculate the final value. However, for a definite integral, such limits or values become very crucial. Now, there is another thing called endpoint. So, what is it? It is a point at the boundary of either a line segment or a closed interval. However, we will not go into the details of endpoints in this discussion. But the chief question is, what then, is an improper integral or improper integral calculator?

improper integral calculator

Now, before we go into the details of the improper integral calculator, let us see first what it exactly is. So, when you are conducting mathematical analysis, an improper integral is practically the limit of a definite integral. Therefore, this acts as an endpoint of the intervals of integration approaches either a specified real number or positive or negative infinity. Moreover, in some cases, both endpoints can approach limits. So, we often write an integral like this symbolically just like a standard definite integral. However, in some cases, we use infinity as a limit of integration. So now that is a lot of information and it is only natural that you will get confused easily. Therefore, let us see how to work with them. So, in the next section, you will see in detail the working of an improper integral calculator.

Improper integral calculator meaning

So, we all have a rough idea of what an improper integral is. Now the question is why do we need an improper integral calculator? Well, the answer is very simple. We need it for the same reason why we need a calculator for any other integrals. So, an improper integral calculator integrates the integral that you feed into it along with other data and finds out the final integrated value. Moreover, as you might already know, calculus is not at all something easy to do. It is laborious and time-consuming, particularly integration. Therefore, there are high chances of us messing things up in the middle. However, with an improper integral calculator, you can complete the calculation without any hassle in a matter of seconds.

improper integral calculator

In easier terms, an improper integral is something that lies between the definite and the indefinite integral. So, we already know, in an indefinite integral both the limits are positive and negative infinity respectively. On the other hand, definite integrals have fixed limits. However, an improper integral has one definite or fixed limit and the other limit as positive or negative infinity. So, this depends on whether the lower or the upper limit is indefinite. Therefore, the calculation is slightly different in this case. However, this is not the proper way to understand it and is just a simple way to get the thing when you are starting.

Limit of improper integral calculator

So now the limit of an improper integral calculator becomes very important. Moreover, this brings us to the fact there are two kinds of improper integral calculators. Let us take a quick look at them first. So one kind is when the limits of the integration are infinite. On the other hand, the other case is when the integrand becomes infinite but while staying inside the interval of integration. So, you cannot calculate an improper integral if you use a normal Riemann integral. Therefore, this is why you need to know the working of the improper integral calculator. We will come at it presently but first, we need to clarify another basic thing.

Value of improper integral calculator

So, the final integrated value that the improper integral calculator will provide to you depends directly on the limits that you have fed in. Therefore, it is impossible in most cases to get a finite value if none of your limits are finite. So, the understanding of limits in this entire concept becomes crucial. Moreover, once you feed in the integral, depending on the value, the improper integral calculator can also say whether there is a divergence or a convergence. Now, what a divergence or a convergence is, we will see later. So, before that, we need to know how to handle an improper integral calculator in general and where to navigate one on the internet.

Improper integral calculator online

So, the internet has made a lot of things very easy. Math is just one of them. Previously, calculus used to be a tedious manual process and if you did not have a good brain, you would be ruined. But no more, because the internet is a great equalizer. If you can get hold of the perfect improper integral calculator online, you can easily understand the concept and start solving it on your own. There are a lot of calculators out there. However, we have chosen one of the best ones and will shape our discussion as per that. But none of these are free if you need the complete stepwise process.

Improper integral calculator Symbolab

So, Symbolab produces some of the best calculators that you can have on the internet. However, this does not include only calculus calculators. Anyway, in the case of improper integral calculators, it is no different. As against a lot of other calculators, in this improper integral calculator, you can see the entire working out of the solution step-by-step. Therefore, this has various benefits. First, if you are a beginner you have a clear understanding of how to solve a sum because the answer alone cannot help you much. Secondly, if you have made a mistake somewhere, you can trace where that happened and rectify it accordingly. Moreover, you can just run the sum to see if you have done it correctly.

Moreover, in integration, it often happens that there are a number of possible answers to one sum. So, this happens because it depends on the process that you are following. Therefore, it might be so that you have an answer that is different from the one that the improper integral calculator has given. So, you can actually feed in your answer and check whether yours is right. This is a great benefit. Let us now see the stepwise working of the calculator.

Improper integral calculator with steps

If you are a beginner and do not have questions at hand, the improper integral calculator also has the option of examples. So, you can simply click one and check out the entire solution.

Step 1 to use an improper integral calculator.

Go to the Symbolab calculator page. Click on Symbolab and it will directly take you to the required page. So, once you reach there, you will see a toolbox right at the top of the page that contains an assortment of functions. Right below is a search-engine-like box where you have to enter the data. Below this, the segment of the solution comes up once you have fed the data to the improper integral calculator.

Step 2 to use an improper integral calculator.

So in the box that looks like a search engine, there is another smaller box that is basically the space for the integral’s limits. So, you enter the integral with the proper limits. Let us say, in this case, the integral is 0∞(1/x) dx. Therefore, enter the value here, and then you will find a red button right beside that reads “Go”. So, click it for the function to run.

Step 3 to use an improper integral calculator.

So, the improper integral calculator will take a few seconds to calculate. Then in the solutions section, you will find the needed answer. Moreover, you can get the entire stepwise process by clicking the respective option on the right top of the calculator. So, in this case as per the laws of integration, (1/x) dx gives ln IxI. Moreover, the limits are infinity and 0. Therefore, ln I∞I – ln 0 + C where C is the constant. In this case, the entire answer remains undefined since both the logarithms of infinity and 0 are defined. Therefore, it is not a very plausible case.

Step 4 to use an improper integral calculator.

Now, you can also check the boundaries with the help of the improper integral calculator. In this case, the boundaries diverge. To unlock the steps, in this case, you have to pay.

Step 5 to use an improper integral calculator.

Finally, at the bottom, there is another box in the shape of a search engine. Here you can enter data if you have a different answer or feel that the calculator’s solution is wrong. Moreover, right beside it is another red button that reads verify. Click it and the improper integral calculator will check it.

improper integral calculator

Improper integral calculator divergence.

In very simple terms, if the integration of the improper integral does not exist, then we can say it is divergent or that it diverges. Therefore, as per the equation right above, a final value does not exist. It is undefined. Moreover, it is not very rare when you are doing integration. So, this depends a lot on the limits that decide the boundaries. For example, if it was ln 100 and ln 10 instead of infinity and 0, there would have been a valid answer. This is how the improper integral calculator said that 0∞(1/x) dx shows divergence.

improper integral calculator

Improper integral calculator convergence.

Now again to explain very simply, if the integration of an improper integral exists, we say it is convergent or that it shows convergence. Therefore, if the limits are such that after the integration when you place the values as per the limits, you get a defined answer, and you know that the function is a convergent one. So, it is a very simple process and this is how an improper integral calculator finds out as well.

improper integral calculator

Improper integral calculator with example

Let us take another example for better understanding. Let us say 0∞sin x dx.

So, we all know that as per the laws of integration, integration of sin x is – cos x. Therefore, the final value is – cos x + c where c is the constant. However, we need to put limits. So, this comes to be – cos infinity + cos 0. Now, cos 0 is infinity. However, cos infinity is not possible only because the cosine function lies between -1 and 1. So, the answer is again undefined. Hence, when you enter it into an improper integral calculator, it would say that the function diverges.

improper integral calculator

Improper integral calculator FAQs


How do you tell if an improper integral converges or diverges from the calculator?

Ans. So, a normal improper integral calculator will be able to tell you if the integral converges or diverges. If the limits exist, it converges. On the other hand, if the limits do not exist, it diverges. So, in the case of a divergent function, you will get an undefined answer. That way, you will understand as soon as you solve the integration and get the final value.

Does the integral converge or diverge?

Ans. So, an improper integral can either converge or diverge. Both options are available. However, the factor that helps the improper integral calculator gives the final answer is the limit. It depends on how valid the limits are with respect to the integrated value.

What is an improper integral calculator?

Ans. So, an improper integral calculator is a mathematical device. Therefore, it calculates the final value of the integral and also checks its boundaries. Depending on this, it can say whether the value is defined or not. So, it can say if the function is convergent or divergent.