This example shows how to use the **antiderivative calculator** to integrate sin (x) + x with respect to x, you must enter:

- antiderivative(`sin(x)+x;x`) or
- antiderivative(`sin(x)+x`), when there is no ambiguity about the variable of integration.

The antiderivative calculator allows to calculate an antiderivative online with detail and calculation steps.

The **antiderivative calculator** allows to find antiderivative function, antiderivative integral or indefinite integral of a function using
integration properties and different calculation mechanisms online. The antiderivative calculator is able to do symbolic **antidifferentiation**.

The **inverse derivative calculator** allows to :

- Calculate one of antiderivatives of a polynomial
- Calculate antiderivatives of the usual functions
- Calculate antiderivatives of a addition
- Calculate antiderivatives of a subtraction
- Calculate antiderivatives of a rational fraction
- Calculate antiderivatives of composed functions
- Calculate a antiderivative using integration by parts
- Calculate a primitive using the table of usual antiderivatives

The antiderivative calculator allows to **integrate online** any polynomial.

For example, **to compute an antiderivative** of the polynomial following `x^3+3x+1`, you must enter
antiderivative(`x^3+3x+1;x`), after calculating the result
`(3*x^2)/2+(x^4)/4+x` is returned.

The antiderivative calculator is able **to calculate online** all **antiderivatives** of **usual functions** : sin, cos, tan, ln, exp, sh, th, sqrt (square root), and many more ...

So, to obtain an antiderivative of the cosine function with respect to the variable x, type, antiderivative(`cos(x);x`), result `sin(x)` is returned after calculation..

Integration is a linear function, using this property allows the function to get the required result.

For the **online calculation** of **antiderivative** of **function sum**,
simply type the mathematical expression that contains the sum, specify the variable and apply function .

For example, to calculate an antiderivative line of the sum of the following functions `cos(x)+sin(x)`type antiderivative(`cos(x)+sin(x);x`), after calculating the result `sin(x)-cos(x)` is returned.

**To calculate** **online** an **antiderivative** of a **difference** of
**function**, just input mathematical expression that contains the difference, specify variable and
apply function antiderivative.

For example, to calculate online an antiderivative of the difference of the following functions `cos(x)-2x` type antiderivative(`cos(x)-2x;x`), after calculating the result `sin(x)-x^2` is displayed.

To **find the antiderivative of a rational fraction**, the calculator will use its
decomposition into simple elements.

For example, to find a antiderivative of the following rational fraction `(1+x+x^2)/x` : you must enter antiderivative(`(1+x+x^2)/x;x`)

**To calculate** **online** an **antiderivative** of **composition of functions** of
the form u(ax+b), where u is a usual function, simply type mathematical expression that contains the function,
specify variable and apply function antiderivative.

For example, to calculate online an antiderivative of the following function `exp(2x+1)` you must enter antiderivative(`exp(2x+1);x`), after calculating the result `exp(2x+1)/2` is displayed.

For example, to calculate online an antiderivative of the following function `sin(2x+1)` you must enter antiderivative(`sin(2x+1);x`), to get the following result `-cos(2*x+1)/2`.

For calculation of some functions, calculator is able to use **integration by parts**. The formula used is as follows :
Let f and g be two continuous functions, `int(f'g)=fg-int(fg')`

For example, to calculate an antiderivative `x*sin(x)`, calculator uses the integration by parts, to get the result, you must enter antiderivative(`x*sin(x);x`), after calculation, result sin(x)-x*cos(x) is returned with steps and detailed calculations.

To **integrate a function**, the following **formulas** can be used and the usual **calculation rules** applied:

antiderivative(`k;x`) | `kx + c` |

antiderivative(`x`) | `x^2/2 + c` |

antiderivative(`x^n`) | `x^(n+1)/(n+1) + c` |

antiderivative(`1/x^n`) | `-1/((n-1)*x^(n-1)) + c` |

antiderivative(`abs(x)`) | `x/2 + c` |

antiderivative(`"arccos"(x)`) | `x*arccos(x)-sqrt(1-(x)^2) + c` |

antiderivative(`"arcsin"(x)`) | `x*arcsin(x)+sqrt(1-(x)^2) + c` |

antiderivative(`"arctan"(x)`) | `x*arctan(x)-1/2*ln(1+(x)^2) + c` |

antiderivative(`"ch"(x)`) | `sh(x) + c` |

antiderivative(`cos(x)`) | `sin(x) + c` |

antiderivative(`"cotan"(x)`) | `ln(sin(x)) + c` |

antiderivative(`"coth"(x)`) | `ln(sh(x)) + c` |

antiderivative(`exp(x)`) | `exp(x) + c` |

antiderivative(`ln(x)`) | `x*ln(x)-x + c` |

antiderivative(`log(x)`) | `(x*log(x)-x)/ln(10) + c` |

antiderivative(`"sh"(x)`) | `ch(x) + c` |

antiderivative(`sin(x)`) | `-cos(x) + c` |

antiderivative(`sqrt(x)`) | `2/3*(x)^(3/2) + c` |

antiderivative(`tan(x)`) | `-ln(cos(x)) + c` |

antiderivative(`"th"(x)`) | `ln(ch(x)) + c` |

By applying the integration formulas and using the **table of usual antiderivatives**, it is possible to calculate many function antiderivatives integral.
These are the calculation methods used by the calculator to find the indefinite integral.

To practice the different calculation techniques, several quizzes on the calculation of an antiderivative are proposed.

antiderivative(fonction;variable), function is the function to integrate.

This example shows how to use the **antiderivative calculator** to integrate sin (x) + x with respect to x, you must enter:

- antiderivative(`sin(x)+x;x`) or
- antiderivative(`sin(x)+x`), when there is no ambiguity about the variable of integration.

See also

- Even or odd function calculator : is_odd_or_even_function. Calculator for determining whether a function is an even function and an odd function.
- Partial fraction decomposition calculator : partial_fraction_decomposition. The calculator allows a rational fraction to be broken down into simple elements.
- Derivative calculator : derivative. The derivative calculator allows steps by steps calculation of the derivative of a function with respect to a variable.
- Taylor expansion calculator : taylor_series_expansion. The taylor series calculator allows to calculate the Taylor expansion of a function.
- Properties of a numerical function : study_function. This calculator lets you study a function using the following algebraic calculation tools: derivative, antiderivative, is_odd_or_even_function, equation_tangent_line, discriminant, degree, valuation.
- Integral calculator : integral. The integral calculator calculates online the integral of a function between two values, the result is given in exact or approximated form.
- Indefinite integral calculator : antiderivative. The antiderivative calculator allows to calculate an antiderivative online with detail and calculation steps.
- Limit calculator : limit. The limit calculator allows the calculation of the limit of a function with the detail and the calculation steps.