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