Definite Integral Calculator

The definite integral calculator helps to calculate the definite integral of different integral function and gives solution in just a few seconds.

with respect to

Upper Bound

Lower Bound

Feedback

How easy was it to use our calculator? Did you face any problem tell us!

Get the Widget

Add this powerfull tool to your website and help them balance chemical equations in second

Introduction to the Definite Integral Calculator:

Definite integral calculator with steps is the best online source that helps you to evaluate the definite integral problems in less than a minute. It is used to find the area under a curve in a graph over a bounded region.

Definite Integral Calculator with steps

Definite integral becomes complicated when you try to solve it manually. You may get stuck or you do not get the accurate solution to your given problems. That is why we introduce the integral calculator with limits, you just need to put your question and it will provide you accurate solution in just one click.

What is a Definite Integral?

Definite integral is a basic concept in calculus that expresses the area under a closed surface between two specific points along the x-axis. It is denoted as:

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

f(x) is the intregand function of x.
dx is the integration variable.
a and b are the two endpoints (lower and upper limits respectively).

$$ \int_a^b f(x)\; dx \;=\; F(b) - F(a) $$

The definite integral calculator gives output in the form F(b) and F(a) after applying upper limit b and lower limit a respectively.

How to Calculate a Definite Integral?

For the calculation of definite integral problems, the definite integration calculator uses the basic rules of integration so that you do not face any difficulty during evaluation. Nonetheless, let's see the process of calculating definite integral stepwise for complete clarity.

Step 1:

Identify the given integral function and its integration variable.

Step 2:

After analyzing the given function, apply the basic formula of integration for a direct solution.

Step 3:

If it cannot be solved by direct integral rules then use these methods like integration by partial fraction method, integration by parts, trigonometric substitution method, etc to find a solution.

Step 4:

Apply both the upper and the lower limit in the above result, and simplify it.

Step 5:

Now, you get the solution to a given definite integral problem.

Solved Example of Definite Integral:

The solved example of definite integral will give you complete details about how the definite integral calculator with steps evaluate the definite integral problems.

Example:

$$ \int_1^3 x^2 dx $$

Solution:

Determine the given definite integral function,

$$ \int_1^3 x^2 dx $$

Apply the power rule of integration to solve it with respect to x,

$$ \int_1^3 x^2 dx \;=\; \biggr[ \frac{x^3}{3} \biggr]_1^3 $$

Apply the upper and the lower limit,

$$ \int_1^3 x^2 dx \;=\; \biggr[\frac{x^3}{3} \biggr]_1^3 $$

$$ =\; \frac{3^3}{3} - \frac{1^3}{3} \;=\; \frac{27}{3} - \frac{1}{3} $$

$$ =\; 9 - \frac{1}{3} \;=\; \frac{26}{3} $$

Therefore, the given definite integral problem solution is 26/3.

How to Use the Definite Integral Calculator?

The integral calculator with limits has a user-friendly interface, you just need to give your input in this tool to get solution without doing anything else. Follow our guidelines before using it. These guidelines are:

Enter the function f(x) of the definite integral that you want to evaluate in the input box.
Select the integration of the variable for definite integral in the input box.
Add the upper limit b and lower limit a in its respective input box.
Check the given definite integral value before hitting the calculate button to start the evaluation process in the definite integration calculator.
The “Calculate” button gives you the solution of your given definite integral problem
If you want to try our definite integral solver for the first time then it must be suggested to try out the load example to check its working process.
The “Recalculate” button of definite integrals calculator brings back to a new page for finding more example solutions of definite integral problems.

Output from Integral Calculator with Limits:

Definite Integral Calculator gives you the solution to a given definite integral question when you add the input into it. It may contain as:

Result Option:

When you click on the result option of the integration calculator with limits it gives you a solution to the definite integral problem.

Possible Steps:

It provides you with a solution to the definite integral problem where calculation steps are included.

Useful Features of Definite Integration Calculator:

The integral calculator with bounds provides you with many useful features that you get when you calculate the definite integral problems and provides solutions. These features are:

The integral calculator with limits is a free tool that enables you to evaluate the definite integral problems freely.
It is a manageable tool that can solve different types of functions like logarithmic, exponential, and trigonometric to find the solution of definite integral.
Our definite integral solver helps you to get a strong hold on the definite integral concept when you use it for practice.
It saves the time that you consume on the calculation of the definite integral problem solution in a couple of minutes.
The definite integrals calculator provides exact solutions as per your input value when you use it to calculate the definite integral problems without any error.
Definite integral calculator with steps is an educational tool that helps you to teach your children, and students in a simple and fun way.

Related References

How would you explain definite integral?
Give an example of definite integral in steps:
How to evaluate definite integral?
Explain the formula of definite integral:

Frequently Ask Questions

What is the difference between definite and indefinite integrals

Definite and indefinite integrals are two concepts in calculus that involve integration, but they have different purposes and distinct characteristics.

Indefinite Integral:

An indefinite integral is expressed as a family of functions whose derivative is the integrand. It does not have specified limits and a constant of integration. The notation for an indefinite integral is ∫ f(x)dx.

Indefinite integrals do not have upper and lower limits of integration. It includes a constant C. To solve the indefinite integral of a function f(x), the formula is written as ∫ f(x) dx = F(x) + C where F(x) is an antiderivative of f(x).

Definite Integral:

A definite integral represents the total area under the curve of a function between two specific points on the x-axis. It has specified upper and lower limit values. The notation for a definite integral is ∫a^b f(x) dx. Definite integrals have specified lower (a) and upper (b) limits.

The result of a definite integral represents the net area under the curve y = f(x) from x = a to x = b. The definite integral is calculated using the antiderivative F(x) of f(x) is ∫a^b f(x) dx = F(b) − F(a).

Can a definite integral be negative

Yes, a definite integral can be negative. The value of a definite integral can be negative, positive, or zero depending on the behavior of the integrand f(x) over a closed interval [a,b].

When a Definite Integral is Negative

Integrand below the X-Axis:

If the function f(x) is negative (below the x-axis) over the entire interval [a,b], the definite integral will be negative. It represents the areas below the x-axis that are considered negative otherwise if it is above the x-axis it is positive.

Integrand below or above the X-Axis:

If the function f(x) is both above and below the x-axis over the interval [a,b], the definite integral will be the total area, which can be negative if the area below the x-axis exceeds the area above the x-axis.

Can you use integration by parts for definite integrals

Yes, you can use integration by parts for definite integrals. The formula for finding integration by parts is:

$$ \int u\; dv \;=\; uv − \int v\; du $$

For definite integrals, the formula is adjusted for the limits of integration:

$$ \int_a^b u\; dv \;=\; uv ∣_a^b − \int_a^b v\; du $$

This means you first compute uv and then evaluate its limits a and b, and subtract the integral of vdu over the same limits.

How to calculate definite integral from 0 to infinity

To calculate a definite integral from 0 to infinity, you need to consider the definite integral as an improper integral and use the limit process to evaluate it. The process is explained with example,

$$ \int_0^∞ f(x)\; dx \;=\; \lim_{b \to ∞} \int_0^b f(x)\; dx $$

Example:

$$ \int_0^∞ e^-x dx $$

Solution:

To convert a definite integral into an improper integral, apply a limit such as,

$$ \int_0^∞ e^{-x}\; dx \;=\; \lim_{b \to ∞} \int_0^b e^{-x} dx $$

Evaluate the integral and apply upper and lower limits a = 0, b = b

$$ \int_0^b e^-x\; dx \;=\; [-e{^-x}]_0^b \;=\; -e^{-b} - (-e^0) \;=\; -e^{-b} + 1 $$

Then take the limit of the above result of the definite integral,

$$ \lim_{b \to ∞} (-e^{-b} + 1) \;=\; 0 + 1 \;=\; 1 $$

Because e^b approaches infinity it is equal to zero. Therefore the solution to the improper integral is,

$$ \int_0^∞ e^{-x} dx \;=\; 1 $$

How to calculate a definite integral of 1 xlnx

To calculate the definite integral of the function xln⁡(x), use the method of integration by parts. Integration by parts is based on the formula:

$$ \int u\; dv \;=\; uv - \int v\; du $$

Find the value of u and v , du, dv

$$ u \;=\; ln(x) $$

$$ dv \;=\; x\; dx $$

$$ du \;=\; \frac{d}{dx} (ln(x)) dx \;=\; \frac{1}{x} dx $$

$$ v \;=\; \int x\; dx \;=\; \frac{x^2}{2} $$

Put these values in the above formula,

$$ \int x\; ln (x)\; dx \;=\; ln(x) . \frac{x^2}{2} - \int \frac{x^2}{2} . \frac{1}{x} dx $$

$$ =\; \frac{x^2 ln(x)}{2} - \int \frac{x^2}{2x} dx $$

$$ =\; \frac{x^2 ln(x)}{2} - \int \frac{x}{2} dx $$

$$ =\; \frac{x^2 ln (x)}{2} - \frac{1}{2} \int x dx $$

$$ =\; \frac{x^2 ln (x)}{2} - \frac{1}{2} . \frac{x^2}{3} $$

$$ =\; \frac{x^2 ln (x)}{2} - \frac{x^2}{4} $$

Apply both limits as,

$$ \int_a^b x\; ln(x)\; dx \;=\; \left[\frac{x^2 ln(x)}{2} - \frac{x^2}{4} \right]_a^b $$

$$ \int_a^b x\; ln (x)\; dx \;=\; \left(\frac{b^2 ln (b)}{2} - \frac{b^2}{4} \right) - \left( \frac{a^2 ln(a)}{2} - \frac{a^2}{4} \right) $$

Amanda Jepson

Last Updated February 21, 2024