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 of 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 the interval of integration is 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 converge otherwise it diverges because of the involvment of infinity.
Types of Improper Integral:
There are three types of improper integral functions, which gives solutions only if you 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 it will converge, if the limit exists it will 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 uses a simple mechanism for solving the complex improper integral function as it has an updated algorithm that can solve all types of improper integral functions without 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,
$$ =\; \lim_{t \to -\infty} \left( \frac{1}{2} tan^{-1} 0 -\frac{1}{2} tan^{-1} \frac{t}{2} $$
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 process of our improper integral calculator.
Example: Evaluate,
$$ \int_{-\infty}^{+\infty} xe^x dx $$
Solution:
Identify the given integral function and check the type of improper integral function,
$$ \int_{-\infty}^{+\infty} xe^x dx \;=\; \int_{-\infty}^0 xe^x\; dx + \int_0^{+\infty} xe^x\; dx $$
Write the given improper integral into the function mentioned above,
$$ \int_{-\infty}^0 xe^x\; dx \;or\; \int_0^{+\infty} xe^x\; dx $$
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 the 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 the convergent or divergent integral calculator then use the load example option.
- The “Recalculate” button allows you to evaluate more questions of improper integral.
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 of the improper integral problem.
Steps Box:
Click on the steps option so that you get the solution of improper integral questions in a steps.
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 questions or examples.
Our convergent integral calculator only takes the input value and gives the solution of improper integral. These benefits makes our tool unique and excite users to use it for evaluation. The benefits are:
- It is a trustworthy tool as it always provides you with accurate solutions of 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 to learn about the concept of improper integral problems very easily on online platforms.
- 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 do not take any fee and allows you to get the solution of improper integral problems.
- It is an easy-to-use tool, anyone even a beginner can easily use it to get the solution of improper integral problems
- Improper integral calculator can be opened on a desktop, mobile, or laptop through the internet to solve complex improper integral problems.
