Tool to find primitives of a function. Integration of a function is the calculation of all its primitives, the inverse of the derivative.

Primitives Functions - dCode

Tag(s) : Functions, Symbolic Computation

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Tool to find primitives of a function. Integration of a function is the calculation of all its primitives, the inverse of the derivative.

The primitive of a function \( f \) defined over an interval \( I \) is a function \( F \), defined and differentiable over \( I \), which derivative is \( f \), ie. \( F'(x) = f(x) \).

Example: If \( f(x) = x^2+sin(x) \) then the primitive is \( F(x) = \frac{1}{3}x^3-cos(x) + C \) (with \( C \) a constant).

dCode knows all functions and their primitives. Enter the function and its variable to integrate and dCode do the computation of the primitive function.

Mathematicians talks about finding the function calculating the area under the curve.

Function | Primitive |
---|---|

$$ \int \,\rm dx$$ | $$x + C$$ |

$$ \int x^n\,\rm dx$$ | $$ \frac{x^{n+1}}{n+1} + C \qquad n \ne -1 $$ |

$$ \int \frac{1}{x}\,\rm dx$$ | $$\ln \left| x \right| + C \qquad x \ne 0 $$ |

$$ \int \frac{1}{x-a} \, \rm dx $$ | $$\ln | x-a | + C \qquad x \ne a $$ |

$$ \int \frac{1}{(x-a)^n} \, \rm dx$$ | $$-\frac{1}{(n-1)(x-a)^{n-1}} + C \qquad n \ne 1 , x \ne a $$ |

$$ \int \frac{1}{1+x^2} \, \rm dx$$ | $$\operatorname{arctan}(x) + C $$ |

$$ \int \frac{1}{a^2+x^2} \, \rm dx$$ | $$\frac{1}{a}\operatorname{arctan}{ \left( \frac{x}{a} \right) } + C \qquad a \ne 0 $$ |

$$ \int \frac{1}{1-x^2} \, \rm dx$$ | $$\frac{1}{2} \ln { \left| \frac{x+1}{x-1} \right| } + C $$ |

$$ \int \ln (x)\,\rm dx$$ | $$x \ln (x) - x + C $$ |

$$ \int \log_b (x)\,\rm dx$$ | $$x \log_b (x) - x \log_b (e) + C $$ |

$$ \int e^x\,\rm dx$$ | $$e^x + C $$ |

$$ \int a^x\,\rm dx$$ | $$\frac{a^x}{\ln (a)} + C \qquad a > 0 , a \ne 1 $$ |

$$ \int {1 \over \sqrt{1-x^2}} \, \rm dx$$ | $$\operatorname{arcsin} (x) + C $$ |

$$ \int {-1 \over \sqrt{1-x^2}} \, \rm dx$$ | $$\operatorname{arccos} (x) + C $$ |

$$ \int {x \over \sqrt{x^2-1}} \, \rm dx$$ | $$\sqrt{x^2-1} + C $$ |

$$ \int \sin(x)\,\rm dx $$ | $$ -\cos(x)+C $$ |

$$ \int \cos(x)\,\rm dx $$ | $$ \sin(x)+C $$ |

$$ \int \tan(x)\,\rm dx $$ | $$ -\ln|\cos(x)|+C $$ |

Some complex functions can involve the functions \( F \) and \( E \) respectively first and second kind of elliptic integrals.

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Source : https://www.dcode.fr/primitive-integral

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