Integral Calculator

Calcualte the integral for any function for free!

This tool will calculate the integral of the inputed equation.
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How to Calculate an Antiderivative Using Our Indefinite Integral Calculator

To calculate an antiderivative using our indefinite integral calculator, follow these simple steps:

  • Input the Equation: Enter the function you want to integrate in the designated field. Ensure you use the correct mathematical notation.
  • Specify the Variable: Indicate the variable with respect to which you are integrating, (typically x).
  • Click 'Calculate': Once your equation and variable are entered, click the 'Calculate' button.
  • View the Result: The antiderivative calculator will display the integral of the given function.

Remember, the antiderivative is a function whose derivative is the original function, and it is represented as the integral of the function.

How to Integrate an Equation Using Integral Formulas

Integrating an equation involves following specific integral formulas. Here are the basic steps:

  • Identify the Type of Function: Determine if the function is a polynomial, trigonometric, exponential, or another type.
  • Select the Appropriate Formula (listed below): Use the corresponding integral formula for the function type. For example, the integral of ax^n is (ax^(n+1))/(n+1) when n does not equal -1.
  • Apply the Formula: Substitute your function into the chosen formula.
  • Add the Constant of Integration: Don't forget to add '+ C' to represent the constant of integration.

What is an Indefinite Integral

An indefinite integral, often simply called an integral, refers to the antiderivative of a function. It is a central concept in calculus that represents the generalization of the area under a curve. When you calculate an indefinite integral, you are essentially finding all the functions that have a given function as their derivative.

  • General Form: An indefinite integral is expressed as ∫ f(x) dx, where f(x) is the function to be integrated, and 'dx' signifies integration with respect to the variable x.
  • Constant of Integration: Since derivatives of constant terms are zero, the indefinite integral includes a '+ C' to account for any constant that was differentiated to get the original function.
  • Reversing Differentiation: The process of integration can be thought of as the reverse of differentiation. If differentiation gives the rate of change, integration sums up all these small changes to give the total value.
  • Applications: Indefinite integrals have wide applications in calculus, physics, engineering, and economics, often used to find quantities like displacement, area, volume, and total profit or cost.

Integral Formulas List

Here's an expanded list of common integral formulas to assist you when performing integration:

  • Power Rule: ∫ xn dx = xn+1/(n+1) + C when n does not equal -1
  • Exponential Function: ∫ ex dx = ex + C
  • Natural Logarithm: ∫ ln x dx = x ln x - x + C
  • Trigonometric Functions:
    • ∫ sin x dx = -cos x + C
    • ∫ cos x dx = sin x + C
    • ∫ tan x dx = -ln|cos x| + C
    • ∫ sec2 x dx = tan x + C
    • ∫ csc x cot x dx = -csc x + C
    • ∫ sec x tan x dx = sec x + C
    • ∫ csc2 x dx = -cot x + C
  • Inverse Trigonometric Functions:
    • ∫ 1/√(1-x2) dx = arcsin x + C
    • ∫ -1/√(1-x2) dx = arccos x + C
    • ∫ 1/(1+x2) dx = arctan x + C
  • Hyperbolic Functions:
    • ∫ sinh x dx = cosh x + C
    • ∫ cosh x dx = sinh x + C
  • Integration by Parts (Formula):
    • ∫ u dv = uv - ∫ v du

Good luck, and don't forget to bookmark this integral calculator when you need to convert a function to its integrated form.