Integral Calculator: Definite integral and Indefinite Integral (Antiderivative Calculator)

Online Integral Calculator


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∫

   
Use inf for +∞ and -inf for -∞








Solution:

Integral calculus is one of the most important areas in the world of mathematics, for this reason we put this online Integral Calculator in your hands. With this Integral Solver you will be able to calculate all kinds of integrals thanks to the fact that it uses a powerful mathematical processor. With the push of a button you can convert it from a Definite Integral Calculator to an Indefinite Integral Calculator and vice versa.

So that you can obtain the maximum potential of the integral calculator, below you will find the instructions section and a little further down a summary of the main theoretical concepts related to the calculation of integrals.

Instructions for using the Integral Calculator

Integral Solver - Instructions for use

To use the calculator follow these steps:

  1. Choose the calculation mode you want to use: to calculate definite integrals press the “Definite” button and to solve indefinite integrals press the “Indefinite” button.
  2. Choose the differential for the integral you want to calculate. You must do this taking into account the independent variable of the function that you will enter in the calculator. For example if you want to integrate f(x)=x2+1 you must choose the differential dx.
  3. Write the function in the main input, for this you must make use of the list of valid functions that are presented after step 4. If you have chosen the Definite Integral Calculator mode, you must also enter the limits of integration. If you want to compute the definite integral over an infinite interval, you would write inf for +∞ and -inf for –∞.
  4. Once you have completed the previous steps, you just have to press the “Calculate” button, by doing so a box with the solution will be displayed automatically.
Valid functions and symbols Description
sqrt() Square root
ln() Natural logarithm
log() Logarithm base 10
^ Exponents
abs() Absolute value
sin(), cos(), tan(), csc(), sec(), cot() Basic trigonometric functions
asin(), acos(), atan(), acsc(), asec(), acot() Inverse trigonometric functions
sinh(), cosh(), tanh(), csch(), sech(), coth() Hyperbolic functions
asinh(), acosh(), atanh(), acsch(), asech(), acoth() Inverse hyperbolic functions
pi PI number (π = 3.14159...)
e Neper number (e= 2.71828...)
i To indicate the imaginary component of a complex number.
inf ∞

Integral definition

The integral is the mathematical method that allows obtaining the primitive function F(x) from a function f(x) that has been previously derived. That is, the integral is the opposite operation of the derivative just as multiplication is to division. For this reason, integral is also called antiderivative.

If F'(x)=f(x),

⌠⌡f(x) dx = F(x)+C

where,

  • ∫ is the integral symbol.
  • ∫f(x)d(x) is called the indefinite integral of f(x) with respect to x.
  • The function f(x) is called the integrand, and this mathematical operation is called integration.
  • d(x) is called the differential of x.
  • F(x) is the primitive or antiderivative function and C is the constant of integration.

What is an Indefinite Integral?

From what was explained above we can conclude that the function f(x) has infinite primitives, since if F(x) is primitive of f (x), so will any other function defined as G(x) = F (x) + C, where C is a constant value. The concept of indefinite integral is used to refer to the set of all antiderivatives of a function f(x).

For example, the indefinite integral of  f(x)=2x is  x2+C, which groups the family of primitive functions: x2, x2+1, x2+2, x2+3, …

Indefinite Integral of 2x

Definition of Definite Integral

The definite integral of a function f(x) determines the area under the curve on a closed interval [a, b].

⌠⌡
b
 
a
 f(x) dx

Barrow’s Rule tells us that the definite integral of f(x) on the closed interval [a,b] is equal to the difference between the values that a primitive function F(x) takes on that interval. From this rule we obtain the formula for the definite integral:

⌠⌡
b
 
a
 f(x) dx = F(b)−F(a)

Definite integral formula

Example: Calculate the definite integral of f(x)=x^2+3 in the interval [0, 2]:

Improper integral

An improper integral is a special type of definite integral in which the function becomes undefined at some point in the interval of integration. This may be because one or both limits of integration are infinite, or because there is a point within the interval of integration at which the function does not exist.

There are three types of improper integrals:

  1. Improper integrals of type 1 are those in which one or both limits of integration have an infinite value and the function is continuous on that interval.
Improper integrals of type 1
  1. Improper integrals of type 2 are integrals that experience an asymptotic discontinuity in the interval of integration.
  2. The improper integrals of type 3 are a combination of the previous two.

The Integral Solver that we present to you here is also a wonderful improper integral calculator with which you will be able to solve all kinds of improper integrals in a simple way.

Integration rules

The integration rules are a set of guidelines that help us to perform the integration of basic functions in a simple way. Here are the basic integration rules:

Integral power rule

⌠⌡xn dx = xn+1n+1 +C

Integral of a constant

⌠⌡a dx = ax+C

Integration rule for ex

⌠⌡ex dx = ex+C

Integration rule for ax

⌠⌡ax dx = axln(a) +C

Integral of 1/x

⌠⌡ 1x dx = ln(x)+C

Properties of integrals

⌠⌡f(x)+g(x) dx = ⌠⌡f(x) dx+⌠⌡g(x) dx
⌠⌡f(x)-g(x) dx = ⌠⌡f(x) dx-⌠⌡g(x) dx
⌠⌡k•f(x) dx = k⌠⌡f(x) dx, where k is a constant

Trig integral rules

⌠⌡sin(x) dx = −cos(x)+C
⌠⌡cos(x) dx = sin(x)+C
⌠⌡tan(x) dx = −ln(cos(x))+C
⌠⌡csc(x) dx = ln(tan( x2 ))+C
⌠⌡sec(x) dx = ln(tan( x2 )+ π4 )+C
⌠⌡sec(x) dx = ln(tan( x2 )+ π4 )+C
⌠⌡cot(x) dx = ln(sin(x))+C

Integral table

As a complement to the antiderivative calculator, here we offer you a table with more than 110 integrals. This integral table allows you to enter the values of the coefficients which is very helpful when studying methods of solving integrals.

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