Tool to calculate the integral of a function. The computation of an definite integral over an interval consist in measuring the area under the curve of the function to integrate.

Definite Integral - dCode

Tag(s) : Functions, Symbolic Computation

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This page deals with integral calculation on an interval. For the general case, with infinite bounds, see the calculation of primitives.

Tool to calculate the integral of a function. The computation of an definite integral over an interval consist in measuring the area under the curve of the function to integrate.

To perform an integral calculation (integration), first, calculate the corresponding primitive function.

For a function \( f(x) \) to be integrated over \( [a;b] \) and \( F(x) \) the primitive of \( f(x) \). Then $$ \int^b_a f(x) \mathrm{ dx} = F(b)-F(a) $$

Example: Integrate \( f(x) = x \) over the interval \( [0;1] \). Calculate its primitive \( F(x) = \frac{1}{2} x^2 \) and so integral $$ \int^1_0 f(x) \mathrm {dx} = F(1) - F(0) = \frac{1}{2} $$

Enter the function, its lower and upper bounds and the variable to integrate, dCode will make the computation.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Reminder: the derivative of the primitive of a function is the function itself.

Calculation of some forms of integrals involve special functions such as $ E $ and $ F $ which are elliptic integrals or $ I_0, I_n, J_0, J_n, K_0, K_n $ which are Bessel functions.

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integral,function,integration,integrate,calculus,derivative,antiderivative,primitive

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

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