Tool to find primitives of functions. 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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The primitive (or indefinite integral, or antederivative) of a function $ f $ defined over an interval $ I $ is a function $ F $ (usually noted in uppercase), itself defined and differentiable over $ I $, which derivative is $ f $, ie. $ F'(x) = f(x) $.

__Example:__ The primitive of $ f(x) = x^2+\sin(x) $ is the function $ F(x) = \frac{1}{3}x^3-\cos(x) + C $ (with $ C $ a constant).

Computing the antiderivatives of a function involves finding another function which, when derived, gives the original function.

The easiest way to calculate a function primitive is to know the list of common primitives and apply them.

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

The usual primitives to know: (with $ C $ any constant)

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

constant $$ \int a \, \rm dx $$ | $$ ax + C $$ |

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

hyperbolic sine $$ \int \sinh(x)\,\rm dx $$ | $$ \cosh(x) + C $$ |

hyperbolic cosine $$ \int \cosh(x)\,\rm dx $$ | $$ \sinh(x) + C $$ |

hyperbolic tangent $$ \int \tanh(x)\,\rm dx $$ | $$ \ln(\cosh(x)) + C $$ |

Primitives are useful in many areas of mathematics and physics. Used in conjunction with integration, they solve problems related to the determination of areas under curves, the modeling of continuous phenomena, the analysis of growth and change, as well as the resolution of Differential equations.

The value $ C $ is any constant. The presence of a constant in a primitive has no influence on the value of its derivative. So there are an infinite number of possible primitives, adding $ + C $ does not change the derivative, most of the time taking $ C = 0 $ simplifies the calculations.

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Cite as source (bibliography):

*Primitives Functions* on dCode.fr [online website], retrieved on 2024-06-24,

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

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