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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

### How to calculate a primitive/integral?

The primitive (indefinite integral) 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).

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?

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 $$\int \frac{1}{x} \, \rm dx$$$$\ln \left| x \right| + C \qquad x \ne 0$$
$$\int \frac{1}{x-a} \, \rm dx$$$$\ln | x-a | + C \qquad x \ne a$$
$$\int \frac{1}{(x-a)^n} \, \rm dx$$$$-\frac{1}{(n-1)(x-a)^{n-1}} + C \qquad n \ne 1 , x \ne a$$
$$\int \frac{1}{1+x^2} \, \rm dx$$$$\operatorname{arctan}(x) + C$$
$$\int \frac{1}{a^2+x^2} \, \rm dx$$$$\frac{1}{a}\operatorname{arctan}{ \left( \frac{x}{a} \right) } + C \qquad a \ne 0$$
$$\int \frac{1}{1-x^2} \, \rm dx$$$$\frac{1}{2} \ln { \left| \frac{x+1}{x-1} \right| } + C$$
$$\int \frac{1}{\sqrt{1-x^2}} \, \rm dx$$$$\operatorname{arcsin} (x) + C$$
$$\int \frac{-1}{\sqrt{1-x^2}} \, \rm dx$$$$\operatorname{arccos} (x) + C$$
$$\int \frac{1}{\sqrt{x^2-1}} \, \rm dx$$$$\sqrt{x^2-1} + C$$
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 $$\int \tanh(x)\,\rm dx$$$$-\ln(\cosh(x))+C$$

The primitive calculation of some functions within dCode calculator can involve elliptic integrals, Cosine Integral and Sine Integral, or Spence's function or Euler Beta function.

## Source code

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