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

Antiderivative online

The **antiderivative calculator** allows to find primitive function, primitive 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 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 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:

f(x) | F(x) |
---|---|

k | kx+c |

x | `x^2/2+c` |

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

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

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

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

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

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

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

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

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

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

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

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

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

racine_cubique(x) | `3/4*(x)^(4/3)`+c |

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

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

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

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

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 primitives integral.
These are the calculation methods used by the calculator to find the indefinite integral.

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

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 :

- Antiderivative calculator : antiderivative. The antiderivative calculator allows to calculate a primitive online with detail and calculation steps.
- Derivative calculator : derivative. Derivative calculator allows steps by steps calculation of the derivative of a function with respect to a variable.
- Integral calculator : integral. The integral function calculates online the integral of a function between two values.
- Even or odd function calculator : is_odd_or_even_function. Calculator for determining whether a function is an even function and an odd function.
- Limit of a function : limit. The limit calculator allows the calculation of the limit of a function with the detail and the calculation steps.
- Partial fraction decomposition : partial_fraction_decomposition. The calculator allows a rational fraction to be broken down into simple elements.
- Calculate the Taylor expansion of a function : taylor_series_expansion. The taylor series calculator allows to calculate the Taylor expansion of a function.