# Integral Calculator

\( \)\( \)\( \)\( \)A step by step indefinite integral calculator of functions is presented.

## Definition of Indefinite Integral

The indefinite integral \( \displaystyle \int f(x) dx \) of function \( f(x) \) is given by
\[ \displaystyle \int f(x) dx = F(x) + C \]
such that \( F'(x) = f(x) \).

\( C \) is the constant of integration and \( F(x) \) is called the antiderivative.

Note that once the indefinite integral of \( f(x) \) is calculated, one way to check your answer is to find \( F'(x) \) and compare it to \( f(x) \)

Example

\( \displaystyle \int (x^2 + 1) dx = \dfrac{1}{3} x^3 + x + C\) , hence \( F(x) = \dfrac{1}{3} x^3 + x \)
because \( F'(x) = \dfrac{1}{3} (3x^{3-1} )+1 = x^2 + x \)

## Use of the Indefinite Integral Calculator

1 - Enter and edit function $f(x)$ and click "Enter Function" then check what you have entered.

Note that the five operators used are: + (plus) , - (minus), / (division) , ^ (power) and * (multiplication). (example: f(x) = x^3 - 2*x + 3*cos(3x-3) + e^(-4*x)).(more notes on editing functions are located below)

2 - Click "Calculate Integral" to obain the antiderivative \( \displaystyle F(x) \).

Notes: In editing functions, use the following:

1 - The inverse trigonometric functions are entered as: arcsin() arccos() arctan() and the inverse hyperbolic functions are entered as: arcsinh() arccosh() arctanh()

2 - The five operators used are: + (plus) , - (minus), / (division) , ^ (power) and * (multiplication). (example: f(x) = 2*x^3 + 3*cos(2x - 5) + ln(x))

3 - The function square root function is written as (sqrt). (example: sqrt(x^2-1)

4 - The exponential function is written as (e^x). (Example: e^(2*x+2) )

5 - The log base e function is written as ln(x). (Example: ln(2*x-2) )

Here are some examples of functions that you may copy and paste to practice:

x^2 + 2x - 3 (x^2+2x-1)/(x-1) 1/(x-2) ln(2*x - 2) sqrt(x^2-1)

2*sin(2x-2) e^(2x-3)
1/sqrt(x^2-1) 1/sqrt(1-x^2)

## More References and Links

IntegralsIntegration by Parts

Integration by Substitution

Integration Using Partial Fractions

Table of Integrals