## Primitive Function Calculator

## Answers to Questions

### How to calculate a primitive/integral?

The **primitive** of a function $ f $ defined over an interval $ I $ is a function $ F $ (usually noted in uppercase), 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 use primitive/integration to find the function calculating the area under the curve.

### What is the list of common primitives?

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 $$ |

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

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

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

The **primitive** calculation of some functions within dCode calculator can involve the functions denoted $ F $ and $ E $ respectively first and second kind of elliptic integrals, or $ Li_2 the Spence's function.

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