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Integral calculator

Online integral calculator for mathematical calculations in the Philippines

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Integral allots numbers to functions such that it describes displacement, area (mostly of thin rectangles), volume, and other concepts that come up by combining infinitesimal data. Infinitesimal data, or infinitesimal number, is a quantity that is closer to zero than any standard real number but is not zero.

Integration helps us to find the differential equation of math integrals. Integral functions differentiate and calculate the area under the curve of a given graph.

What is an integral calculator?

In simple terms, an integral calculator is an online calculator or a mathematical tool that makes it easier for us to evaluate the integrals. It provides a very fast and reliable way to solve integral queries. The online integral calculator and the processes it utilizes are different from the inverse derivative calculator, even though these two are the main concepts of integral and differential calculus. It could also be called an integral solver and can be used to solve complicated and non-complicated integration problems.

An online integral calculator helps in defining integrals in order to find the area under the curve like this: f(x) = A

Where:

f(x) = the function,

and A = the area under the curve.

This antiderivative calculating tool is useful for solving definite integrals and indefinite integrals along with proper integrals, improper integrals, double integrals, and even triple integrals.

Moreover, to improve your mathematical intuition and calculus conceptualization, the integration calculator with steps displays graphs which are in different forms, including other important information in its step-by-step solutions.

The integral equations can be very complicated most of the time, even for very basic integrand, and it requires strong algebraic computing skills and capabilities. So, antiderivative calculators with steps employ powerful and large algorithms, which are typically extremely advanced mathematical applications used for solving antiderivative problems.

An integral equation, or the integration formula, is called an integrand; it is always denoted as the function f(x). In the integral calculator, you must enter the value so that it works correctly.

Types of integral calculators

There are two main types of integral calculators: definite integral calculators and indefinite integral calculators. Some calculators have options to choose either of the two, and some are made specifically for either of them.

Definite integral calculators

Definite integrals are the difference between the values of the integral of a function f(x) for an upper bound value b and a lower bound value a of the independent variable, which is x. In other words, this means that there is an interval that is between points [a, b], which means that it is defined.

a and b are called the limits of the definite integral, and are put at the bottom and top of the integration sign, ∫ respectively as shown below:

∫^b a f(x) dx

The defined area here is between a and b on the x-axis of the graph; a is the lower bound value, while b is the upper bound value.

A definite integral calculator could then be said to be a tool that helps in providing a very fast and reliable way to solve definite integrals.

Indefinite integral calculators

Indefinite integrals are integrals which do not have any upper and/or lower limits; they are continuous. Mathematically, if a function f(x) is an anti-derivative of another function f(x), then the most general anti-derivative of f(x) is called an indefinite integral. The mathematical expressions for these are ∫f(x) dx = f(x) + C, where C is the constant of integration.

This could be written as: ∫f(x) dx = f(x)

As seen from the equation above, there is no a and b boundary or limit, which means it is continuous.

An indefinite integral calculator, which could also be known as an indefinite antiderivative calculator, is a tool which helps in providing a fast and reliable way to solve indefinite antiderivative.

What are the benefits of using an integral calculator?

The integral calculator is used by students, teachers, engineers, and researchers. There are many advantages and benefits of using an integral calculator, some of which are:

Provides better accuracy

It is very rampant to make some errors while solving your calculus problems. While solving integration problems by hand, one can sometimes mistake a value or a formula, which can ultimately jinx the answer you get.

Sometimes, the value may be misinterpreted or miscalculated, which can lead to mark deductions in our exams and test papers. The integral calculator is free from any kind of error or miscalculations.

The cutting-edge algorithms incorporated in the step-by-step antiderivative calculator help to receive highly accurate and 100% error-free solutions. The step-by-step solutions from the integral calculator allow us to precisely compare our manually calculated values with the accurate ones.

Saves time

Solving extensive integration problems and finding integrals of a function can be very strenuous and time-consuming; the integration calculations are always very long and need to be accurate from the first to the last steps. So, in order to calculate precise integrals, one must invest a lot of hard work and time.

In contrast, if you use an online integral calculator, your long, hectic, and tiring integrals will be solved within a matter of seconds.

Makes calculations faster

When providing step-by-step solutions to your integral problems, the online calculators for performing integration function problems work very fast. They do not even take a minute to provide you with extensive integral steps solutions to the problems, even showing all the steps.

Within a split of seconds, the sophisticated algorithms of integration calculators solve complex equations fast and accurately. Hence, by being this super-fast, the integral solver benefits us immensely and in no time.

Provides stress-free solutions

The majority of students are highly focused on their studies and get very worried if it is not going well. Calculus itself is a challenging subject, giving them a tough time and some students always end up getting stressed. 

By having efficient, competent, and time-saving integral calculus problem-solving machinery, they will be relieved and have a better understanding too.

When the students are provided with a step-by-step solution along with graphical representations via an integral solver, they will not just be relieved but also have a much better comprehending approach toward their studies when they use the integral calculator.

Is easier to use

Some integral calculators are available online, which makes them freely accessible to all. Moreover, there are no complex commands required to perform an anti-differential calculation. The user interface of the integral calculator is easy to operate, and anyone can use it accurately, even a first-timer.

Being a simple calculating machine which is available online, it can help you avoid the fuss of dealing with complex notations. All these features contribute to access, and it is simple and easy to use.

Functions of an integral calculator

Some of the most useful functions are naturally described by the integration of other known functions. In some instances, these newly defined functions turn out to functions we had already known, maybe in some other context, and sometimes, they are truly new. Some of the functions one may come across in an integration calculator are given below.

∫ f(x) dx - Integral of a function, f with respect to x

f(x) ln ∫ f(x) dx - Integrand

x ln ∫ f(x) dx - Variable of integration

dx ln ∫ f(x) dx - Differentiation goes in the x direction

C - Constant of integration.

Constant of integration, or arbitrary constant

The constant of integration expresses some sense of uncertainty. For example,  given a derivative, there can exist some integrand, which may differ by a set of real numbers. This set of real numbers is represented by the integral constant, C.

Improper integral

Improper integral is when either the lower limit of integration is indefinite, or the upper limit of integration is indefinite, or both the upper and lower limits of integration are indefinite.

Multiple integrals

This means the integral of a function of more than one variable. It is also the integral of a function of one variable over an interval that results in an area, or the double integral of a function of two variables calculated over a region that results in a volume.

Integration techniques

Below are some of the integration techniques used by the integration calculator:

Fundamental theorem

The fundamental theorem states that if one can integrate the integrand, then they can evaluate the definite integral by evaluating the antiderivatives at the endpoints of the interval and then subtracting the constants.

Special functions

The exponential integral Ei is a special function on the complex plane which is defined as one particular definite integral of the ratio between an exponential function and its argument.

How to use a keyboard for online integral calculation?

Here is a quick overview of how to use a keyboard in integral calculators, what each key means, and how one can enter the data correctly. The parsing tree is used to integrate a large number. The following is a list of common functions which you can input and an explanation of what they stand for:

Examples of calculations using an integral calculator

Some examples of integral calculations that could be done using an integral calculator with steps are:

Decomposition

This technique is employed when we can break down the given functions into the sum or difference of smaller functions that we can integrate; the operation can be algebraic, trigonometric, exponential, or a combination of all the three. 

Example: Calculate the integral of the equation (1 + x²)³ dx 

Firstly, it expands the given expression using the formula (a + b)³,

Remember when you expand, (a + b)³  =  a³ + 3a²b + 3ab² + b³

So (1+ x²)³  =  1  + 3x² + 3x⁴ + x⁶

=  ∫dx + 3∫x²dx + 3∫x⁴ dx + ∫ x⁶ dx

=  ∫ dx + 3 ∫(x²) dx + 3 ∫ (x⁴) dx + ∫ x⁶ dx

=  x + 3 (x³/3) + 3 x⁵/5 + x⁷/7 + c

=  x + x³ + (3/5)⁵ x + (1/7) x⁷+ c

This process is done in a flash with the use of a calculator.

Integration by parts 

Supposing our integrand is in the form ∫f(x).g(x)dx. The calculator solves this problem using integration by parts, the formula ∫f(x).g(x) dx = f(x) ∫g(x)dx − ∫ [f′(x) ∫ g(x)dx ]dx is employed. In the graph, the parts could be called the signed area or the unsigned area; the signed areas are usually marked or shaded.

Other examples are integration by substitution and by partial fractions.

Example: Calculate the integral of the equation ∫xsin(x²+5) dx.

To calculate this, we say let u = x²+5, hence du = 2x dx

du = 2xdx ⇒ 1/2du = xdx

When we now substitute and integrate, we have -1/2cos(x²+5) + C.

How to perform complex tasks using an integral calculator?

As said earlier, a parsing tree can be used by the computer to analyze complex mathematical computations. It transforms the f(x) into a form that is better understandable by the computer without any approximation. In doing this, the calculator has to respect the order of operations.

A specific computation mistake in a mathematical expression form is that the multiplication sign can be left out sometimes by the computer; for example, we write "5x" instead of "5*x". The calculator has to detect these cases and insert the multiplication sign.

How to perform complex integration using an online calculator

Once you're done with the above steps and you've clicked on the Calculate button, the calculator will immediately tackle the problem by parts. You'll see your results and everything about it, including the steps.

Example: Evaluate ∫c z¯dz, where C is given by x = 3t, y = t², with − 1 ≤ t ≤ 4.

Here, we are given that C is z(t) = 3t + it²

So therefore, with the function f(z) = z¯,

we have f(z(t)) = 3t + it²

= 3t it².

Also, z′(t) = 3 + 2it, so the integral is:

c z¯dz = ∫^4 −1 (3t it²) (3 + 2it)dt

= ∫^4 −1 (2t³ + 9t + 3t²i)dt

= ∫^4 −1 (2t³ + 9t)dt + i ∫^4 −1 3t²dt

= (1/2t⁴ + 9/2t²) ∣^4 −1 + it³∣^4 −1

= 195 + 65i.