Tool to find primitives of functions. Integration of a function is the calculation of all its primitives, the inverse of the derivative.
Primitives Functions - dCode
Tag(s) : Functions, Symbolic Computation
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Primitives Functions
Answers to Questions (FAQ)
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.
There is no direct formula for calculating a primitive from a function. It is generally possible to express primitives for the usual functions (or combinations of functions) but there is no uniqueness of the primitive (therefore an infinity of solutions) and some primitives cannot be expressed as a combination of usual functions.
What is the list of common primitives?
The usual primitives to know: (with $ C $ any constant)
| Function | Primitive |
|---|---|
| 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 $$ \int \tanh(x)\,\rm dx $$ | $$ \ln(\cosh(x)) + C $$ |
What is the list of primitives for combinations of functions?
Another list to know to calculate primitives of functions combining usual functions and their derivatives
| Function | Primitive |
|---|---|
| fraction of functions $$ \frac{u^{\prime}}{u} $$ | $$ \ln{|u|} $$ |
| exponential of functions $$ \exp(u) \times u^{\prime} $$ | $$ \exp(u) $$ |
| power of functions $$ u^{a} \times u^{\prime} $$ | $$ \frac{u^{a+1}}{a+1} $$ |
| sine of functions $$ \sin(u) \times u^{\prime} $$ | $$ -\cos(u) $$ |
| cosine of functions $$ \cos(u) \times u^{\prime} $$ | $$ \sin(u) $$ |
| logarithm of functions $$ u^{\prime} \times \ln{|u|} $$ | $$ u \times \ln{|u|} -u $$ |
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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