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Integral Calculator

Integral Calculator

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Integral calculator is an online tool that calculates the antiderivative of a function. It works as a definite integral calculator as well as an indefinite integral calculator and lets you solve the integral value in no time.

If you are studying calculus, you may have an idea of how complex integrals and derivatives are. Well, throw away your worries because the integration calculator is here to make your life easier. You can evaluate the integral by only placing the function in our tool.

Now, we will discuss the integral definition, how to use an integral calculator with steps, how to solve integrals with integral solver, and much more.

What is integral?

An integral is the reverse of the derivative. It is as same as the antiderivative. It can be used to determine the area under the curve. Here is the standard definition of integral by Wikipedia.

In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the two main operations of calculus; its inverse operation, differentiation, is the other. 

With an interval of [a, b] of the real line and a real variable x, the definite integral of the given function f can be expressed as:

Generally, there are two types of integrals.

Definite integrals: If the integrals are determined by using lower and upper limit, they are called definite integrals. The standard form of definite integrals can be represented by:

Indefinite integrals: If there is no lower or upper limit defined, the limit is indicated by the integration constant. These types of integrals are called indefinite integrals because there are no limits available.

The standard form of indefinite integrals is:

f(x) dx

How does online integral calculator work?

The antiderivative calculator evaluates a function given by the user and converts it into integration by applying the upper and lower limits in case if it is a definite integral. If it is an indefinite integral, the integrals calculator simply uses the constant of integration to evaluate the expression.

Furthermore, evaluate integral calculator brings a sense of simplicity in calculations of integration by only taking a function from the user. You don’t have to do much other than giving input and this iterated integral calculator does it all on its own, and that too in no time at all.

To use this line integral calculator, follow the below steps:

  • Enter your value in the given input box.
  • Hit the Calculate button to get the integral.
  • Use the Reset button to enter a new value.

Integration by parts calculator will give you a fully evaluated integral function that can be further used in various domains. As mentioned above, integration is the reverse function of derivatives. In case, you need to solve a derivative, use our derivative calculator here.

How to calculate integrals?

Now that you know what integrals are and how can you use the derivative of integral calculator above to solve an integral, you may also want to know how to solve integrals manually. It can be somehow annoying for the ones who are just starting with integrals.

But, don’t worry. We will demonstrate the calculations with examples so that you can grasp it easily. Additionally, you can prepare the topic for your exams using the below guideline.

To calculate integrals, follow the steps below:

  • Determine and write down the function F (x).
  • Take the antiderivative of the function F (x).
  • Calculate the values of upper limit F (a) and lower limit F (b).
  • Calculate the difference of upper limit F (a) and lower limit F (b).

Let’s use an example to understand the method to calculate the definite integral.

Example – Definite integral

For the function f (x) = x – 1, find the definite integral if the interval is [2, 8].

Solution:

Step 1: Determine and write down the function F (x).

F (x) = x – 1, Interval = [2, 8]

Step 2: Take the antiderivative of the function F (x).

F (x) = (x1) dx = (x2 / 2) x

Step 3: Calculate the values of upper limit F (a) and lower limit F (b).

As, a = 1, and b = 10,

F (a) = F (1) = (22 / 2) – 2 = 0

F (b) = F (10) = (82 / 2) – 8 = 24

Step 4: Calculate the difference of upper limit F (a) and lower limit F (b).

F (b) – F (a) = 24 – 0 = 24

This method can be used to evaluate the definite integrals having limits. You can use a double integral calculator above to if you don’t want to indulge in integral calculations.

Example – Integral of a trigonometric function

For the function f (x) = sin (x), find the definite integral if the interval is [0, 2π].

Solution:

Step 1: Determine and write down the function F (x).

F (x) = sin (x), Interval = [0, 2π]

Step 2: Take the antiderivative of the function F (x).

F (x) = sin (x) dx = cos (x)

Step 3: Calculate the values of upper limit F (a) and lower limit F (b).

As, a = 0, and b = ,

F (a) = F (0) = cos (0) = 0

F (b) = F () = cos (2π) = 0

Step 4: Calculate the difference of upper limit F (a) and lower limit F (b).

F (b) – F (a) = 0 – 0 = 0

Along with manual calculation, you can also use our trigonometric substitution calculator above to solve a trigonometric integral in a fraction of seconds.

FAQs

What is an integral calculation?

An integral calculation reverses the function of the derivative by taking the antiderivative of that function. It is used to determine the area under the curve. Integral calculations can be definite if upper and lower limits are there. If there are no intervals, an integral constant C is used and that type of function is called indefinite integral.

What is the derivative of an integral?

If we take the derivative of an integral, both of them will cancel each other because derivative and integral are reverse functions to each other. Integral is the same as antiderivative according to the fundamental theorem of calculus.

Who is the father of integration?

Gottfried Wilhelm Leibniz and Isaac Newton proposed the rules of integration independently at the end of the 17th century. They assumed the integral as an endless sum of rectangles of extremely small width. Bernhard Riemann described integrals in a strict mathematical fashion.

What is the integral of 1?

The integral of 1 is x or x + c because if we add an integral constant. It can be expressed as a diagonal line lies in the 1st and 3rd quadrant of the graph.

1 dx= x + C

What is the integral of sin 2x?

The integral of sin 2x can be calculated by the substitution method. It will be an indefinite integral due to the no interval or upper and lower limits. Here is the integral of sin 2x.

sin (2x) dx= – (1/2) cos (2x) + C