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.

See also: Primitives Functions

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

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