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Primitives Functions

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

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Primitives Functions -

Tag(s) : Functions, Symbolic Computation

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Primitives Functions

Integral Calculator

What is a primitive? (Definition)

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).

How to calculate a primitive/integral?

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.

What is the list of common primitives?

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

FunctionPrimitive
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 $$\int \tanh(x)\,\rm dx$$$$\ln(\cosh(x)) + C$$

Why calculate a primitive?

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.

What does +C mean?

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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Primitives Functions on dCode.fr [online website], retrieved on 2024-06-24, https://www.dcode.fr/primitive-integral

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