Introduction to Improper Integral Calculator:
Improper Integral Calculator is the best online source that helps you evaluate the improper integral of a given function. It is used to find the area under a curve that has two points the upper and the lower points.
Our improper integral convergence calculator is a beneficial tool for students, teachers, and professionals who want to get solutions to improper integral problems for making notes and reports but do not have time to solve them manually.
What is an Improper Integral?
An improper integral is defined as the integral which is continuous over an unbounded region where either the interval of integration is infinite, or infinite. An improper integral is sometimes called a definite function because limits are involved.
An improper integral gives the behavior of a function which means if the limit of a continuous function f(x) exists then the whole function coverage otherwise it diverges because infinity is involved.
Types of Improper Integral:
There are three types of improper integral functions, that cannot give you a solution if you do not split them into two or three. These types are,
- Let f(x) be a continuous function and (+∞, a) is the given limit then you can split them so that the function limit exists. $$ \int_a^{+ \infty} f(x) dx \;=\; \lim_{t \to + \infty} \int_a^t f(x) dx $$
- If f(x) is a continuous function over the (b,-∞) interval then its coverage if the limit exists other its diverge. $$ \int_{- \infty}^{b} f(x) dx \;=\; \lim_{t \to -\infty} \int_t^b f(x) dx $$
- If f(x) is a continuous function over the (+∞,-∞) interval then the given function is mostly diverged because of its limits. $$ \int_{-\infty}^{\infty} f(x)dx \;=\; \int_{-\infty}^0 f(x)dx + \int_0^{+\infty} f(x) dx $$
Calculation Process of Integral Convergence Calculator:
Improper integral calculator has a simple procedure for solving the complex improper integral function because it has an updated algorithm that can solve all types of improper integral functions without any mistakes.
Let's take an example to know how the convergent or divergent integral calculator finds an improper integral in some steps.
Example: Evaluate,
$$ \int_{-\infty}^0 \frac{1}{x^2 + 4} dx $$
Solution:
Step 1:
Identify the given integral function and check which type of improper integral function is,
$$ \int_{-\infty}^0 \frac{1}{x^2 + 4} dx $$
Step 2:
The given improper integral function belongs to type 2. That means infinity is replaced by a constant t and takes the limit such as,
$$ \int_{-\infty}^0 \frac{1}{x^2 + 4} dx \;=\; \lim_{t \to -\infty} \int_t^0 \frac{1}{x^2 + 4} dx $$
Step 3:
Take the integral with respect to x as,
$$ =\; \lim_{t \to -\infty} \frac{1}{2} tan^{-1} \frac{x}{2} \biggr|_t^0 $$
Step 4:
Apply the upper and the lower limits,
$$ =\; \lim_{t \to -\infty} \left( \frac{1}{2} tan^{-1} 0 -\frac{1}{2} tan^{-1} \frac{t}{2} \right) $$
Step 5:
After taking the upper and lower limits, apply the limit value and the solution we get is,
$$ =\; \frac{\pi}{4} $$
Step 6:
The result of the given improper integral function shows that it converges to π/4.
Practical Example of an Improper Integral:
An example of improper integral with the solution is given so that you may understand the working procedure of our improper integral calculator.
Example: Evaluate,
$$ \int_{-\infty}^{+\infty} xe^x dx $$
Solution:
Identify the given integral function and check which type of improper integral function is,
$$ \int_{-\infty}^{+\infty} xe^x dx \;=\; \int_{-\infty}^0 xe^x\; dx + \int_0^{+\infty} xe^x\; dx $$
Write the given improper integral into the above type of function,
$$ \int_{-\infty}^0 xe^x\; dx \;or\; \int_0^{+\infty} xe^x\; dx $$
Integrate both the integral functions one by one,
$$ \int_{-\infty}^{0} xe^x\; dx \;=\; \lim_{t \to -\infty} \int_t^0 xe^x\; dx $$
$$ =\; \lim_{t \to -\infty} (xe^x - e^x) \biggr|_t^0 $$
Apply the upper and lower limits and then apply the limit value. The result is given as,
$$ =\; \lim_{t \to -\infty} (-1 - te^t + e^t) $$
$$ =\; -1 $$
Now take the second integral function and integrate it.
$$ \int_0^{+\infty} xe^x dx \;=\; \lim_{t \to +\infty} \int_0^t xe^x dx $$
$$ =\; \lim_{t \to +\infty} (xe^x - e^x)\biggr|_0^t $$
Apply the upper and lower limits and then apply the limit value. The result is given as,
$$ =\; \lim_{t \to +\infty}(te^t - e^t + 1) $$
$$ =\; \lim_{t \to + \infty}((t - 1)e^t + 1) $$
$$ =\; +\infty $$
Thus, the ∫+∞0 of xex dx diverges. As the ∫+∞-∞ xexdx as well.
How to Use the Improper Integral Calculator?
The improper integral convergence calculator has a user-friendly layout that allows you to solve various types of improper integral problems.
You just need to put your integral problem in this convergent integral calculator and follow some simple steps so you can get results without any inconvenience. These steps are:
- Enter the improper integral function that you want to evaluate in the input box.
- Add the variable of integration in its input box.
- Add the upper limit b and the lower limit a in the input box.
- Recheck your given input value before clicking on the calculate button to get the solution to the improper integral question.
- Click on the “Calculate” button to get the solution to improper integral problems.
- If you want to check the working process behind our convergent or divergent integral calculator then use the load example for the calculation to get an idea about its solution.
- The “Recalculate” button allows you to evaluate more examples of improper integral problem solutions.
Output from Convergent Integral Calculator:
Improper Integral Calculator provides you with an improper integral problem’s solution as per your input when you click on the calculate button. It may include as:
In the Result Box:
When you click on the result button you get the solution to the Improper Integral problem.
Steps Box:
Click on the steps option so that you get the solution of Improper Integral questions in a step-by-step method.
Benefits of Using Convergent or Divergent Integral Calculator:
The integral divergence calculator has multiple benefits that you can avail whenever you use it to solve improper integral problems to get the solution.
Our convergent integral calculator only gets the input value of a and gives an improper integral solution. These benefits will make our tool unique and compel users' attention to use it for evaluation which are given as:
- It is a trustworthy tool as it always provides you with accurate solutions to improper integral problems.
- It is a speedy tool that evaluates improper integral problems with solutions in a couple of seconds.
- The improper integral convergence calculator is a learning tool that helps children about the concept of improper integral problems very easily on online platforms without going to any tutor.
- It is a handy tool that solves improper integral problems quickly because you do not put any type of external effort into the calculation.
- It is a free tool that allows you to use it for the calculation of improper integral problems without taking any fee.
- It is an easy-to-use tool, anyone even a beginner can easily use it for the solution of improper integral problems
- Improper Integral Calculator can open on a desktop, mobile, or laptop through the internet to solve complex improper integral problems.
