Calculator-Online.net

CALCULATOR

ONLINE

Calculator-Online.net

CALCULATOR

ONLINE

Sign In ▾

Follow Us On:

Your Result is copied!
ADVERTISEMENT

Integral Calculator

Enter the mathematical function you want to integrate. For definite integrals, provide the upper and lower bounds.

keyboard

ADVERTISEMENT

Integral Calculator:

Use this integral calculator to compute the integrals of a given mathematical function. Our calculator can handle indefinite integrals (finding the antiderivative of a function) and definite integrals (calculating the numerical value of the integral over a specified interval).

What is An Integral in Calculus?

In mathematics, an integral of a function describes the accumulation of quantities, such as area, displacement, volume, and more, by summing infinite data points. Integration and its inverse operation, differentiation, are the basis for solving problems in physics and mathematics, particularly those involving arbitrary shapes and continuous change.

The process of finding integrals is called integration. The function that is to be integrated is termed the integrand.

For example, in the following equation:

∫3xdx

  • The symbol ∫ represents the integral
  • 3x is the integrand (the function to be integrated)
  • dx indicates the differential of the variable for which we are integrating (x)

For a definite integral over an interval, the result represents the area under the curve of the integrand within those limits. For instance, if A is the area under the curve of f(x) between limits a and b, we will write A = ∫ₐᵇ f(x)dx. 

However, whether you are visualizing areas or solving physics problems, our integral calculator is here to help.

What is the difference between a definite and an indefinite integral?

Indefinite Integrals:

The indefinite integral of the function means to take the antiderivative of that function. This type of integral does not have any upper or lower limit.

Definite Integrals:

The definite integral of the function has the start and end values. Simply, there is an interval [a,b] called the limits, bounds, or boundaries. This type can be defined as the limit of the integral sums when the diameter of the partition tends to zero. 

The following diagram helps to understand the difference between definite and indefinite integrals:

 integral image

Basic Formulas for Integration:

Power Rule:

  • ∫ 1 dx = x + c
  • ∫ xⁿ dx = xⁿ⁺¹ / (n + 1) + c (where n ≠ -1)

Constant Rule:

  • ∫ a dx = ax + c (where a is a constant)

Reciprocal Rule:

  • ∫ (1/x) dx = ln|x| + c

Exponential and Logarithmic Rules:

  • ∫ aˣ dx = aˣ / ln|a| + c (where a > 0 and a ≠ 1)
  • ∫ eˣ dx = eˣ + c

Trigonometric Integrals:

  • ∫ sin(x) dx = -cos(x) + c
  • ∫ cos(x) dx = sin(x) + c
  • ∫ tan(x) dx = - ln|cos(x)| + c
  • ∫ cosec²(x) dx = -cot(x) + c
  • ∫ sec²(x) dx = tan(x) + c
  • ∫ cot(x) dx = ln|sin(x)| + c
  • ∫ sec(x)tan(x) dx = sec(x) + c
  • ∫ cosec(x)cot(x) dx = -cosec(x) + c

Inverse Trigonometric Integrals:

  • ∫ 1 / √(1 - x²) dx = sin⁻¹(x) + c
  • ∫ 1 / √(1 - x²) dx = -cos⁻¹(x) + c
  • ∫ 1 / (1 + x²) dx = tan⁻¹(x) + c
  • ∫ 1 / (1 + x²) dx = -cot⁻¹(x) + c
  • ∫ 1 / (|x|√(x² - 1)) dx = sec⁻¹(x) + c
  • ∫ 1 / (|x|√(x² - 1)) dx = -cosec⁻¹(x) + c

To save yourself from the effort of memorizing all the formulas, simply input your function into our integral solver. It will apply these standardized formulas to compute a definite or an indefinite integral according to the provided input.

How to Solve Integrals Manually (Step-by-Step):

It depends on whether you deal with the indefinite(antiderivative) or definite integral.

Solving Indefinite Integrals:

  1. Determine the function f(x) (the integrand)
  2. Now apply appropriate integration rules (e.g., power rule, constant rule, substitution, integration by parts, trigonometric identities) to evaluate the antiderivative f(x)
  3. The general form of the indefinite integral is F(x) + c, so add the constant of integration, "c."

Example: Solve ∫(x³+5x+6)dx

Solution:

Function: f(x)=x³+5x+6

Apply the power rule:

  • ∫x³dx=x³⁺¹/(3+1)=x⁴/4
  • ∫5xdx=5∫x¹dx=5x¹⁺¹/(1+1)=5x²/2
  • ∫6dx=6x

Add the constant of integration: ∫(x³+5x+6)dx=x⁴+5x²/2+6x+c

To quickly verify your answers step by step and enhance your understanding of integration, try our indefinite integral calculator.

Solving Definite Integrals:

  1. Determine the function f(x) and the limits of integration [a, b]
  2. Find the antiderivative F(x) of the function f(x)
  3. Evaluate the antiderivative at the upper limit
  4. Now, evaluate the antiderivative at the lower limit
  5. Subtract the value at the lower limit from the value at the upper limit

Note: The constant of integration "c" cancels out in definite integrals.

Example: Evaluate the definite integral for ∫sinx dx with an interval of [0, π/2]?

Solution:

Step 1:

Use the formula for trigonometric function: ∫ sinx dx = - cosx + c

Step 2:

Calculate the upper & lower limits for functions f (a) & f (b) respectively:

At the upper limit (b = π/2): F(π/2) = - cos(π/2) = -0 = 0

At the lower limit (a = 0): F(0) = - cos(0) = -1

Step 3:

Calculate the difference between the upper & lower limits:

F(b) - F(a) = 0 - (-1) = 1
 

For complex functions or quick checks, consider using our integration calculator.

How to Use the Integral Calculator?

You can easily calculate the integral of definite and indefinite functions with the assistance of our online integral calculator. Simply, follow these steps:

  • Step #1: Enter the equation that you want to integrate
  • Step #2: Choose the dependent variable involved in the equation
  • Step #3: Select the definite or indefinite integral from the tab
  • Step #4: If you selected the definite option, then you ought to enter the lower & upper bounds or limits in the designated fields
  • Step #5: Once done, click on the “Calculate” button, and the calculator will show the 
    • Definite integral
    • Indefinite integral
    • Complete step-by-step calculations

FAQ’s:

Is Antiderivative the Same as Integral?

These terms are related to each other, but they are slightly different.

  • Antiderivative: The antiderivative of a function f(x) is a function F’(x), i.e, F’(x) = f(x). This means a function can have infinite antiderivatives, differing at a point
  • Indefinite Integral: The integral of a function contains all the antiderivatives of that function 
  • Definite Integral: It represents the signed area under the curve of the function f(x) from x=a to x=b

What are the Applications of Integration in Real Life?

Integration has a wide range of real-life applications, including:

  • Determining the displacement, velocity, and acceleration of an object
  • Calculating the work done by a force on an object 
  • Knowing the center of mass and moment of inertia
  • Analyzing the flow of the fluids
  • Signal processing
  • Calculating the transfer of heat

What is the Significance of the Constant of Integration (+ C) in Indefinite Integrals?

As we know, integration is the reverse of differentiation. The derivative of a constant number is always zero. Therefore, any constant can be present in the original functions. The added “c” indicates the presence of the constant in the original function. The “+C” is used to represent the family of all possible antiderivatives.

How Does a Definite Integral Relate to the Area Under a Curve?

The definite integral is used to find the net area between a curve and the x-axis. With it, you can find the area above and below the x-axis having positive and negative signs, respectively.

References:

From the authorized source of Wikipedia : General understanding about integrals and it's types

From the site of mathisfun : Graph of the integration & integrands, integral notation

From the source of math.com : Basic to advance level formulas for integration

From the site of toppr.com : daily life usage of integration in calculus

animal image

Easter into Action, Save With Satisfaction

UPTO

50 %

OFF

Online Calculator

CALCULATOR

ONLINE

Get the ease of calculating anything from the source of calculator online

© Copyrights 2025 by Calculator-Online.net