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
Primitive Function Calculator
Tool to find primitives of functions. Integration of a function is the calculation of all its primitives, the inverse of the derivative.
Answers to Questions
How to calculate a primitive/integral?
The primitive of a function $ f $ defined over an interval $ I $ is a function $ F $ (usually noted in uppercase), defined and differentiable over $ I $, which derivative is $ f $, ie. $ F'(x) = f(x) $.
Example: If $ f(x) = x^2+sin(x) $ then the primitive is $ 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 |
|---|---|
| $$ \int \,\rm dx $$ | $$ x + C $$ |
| $$ \int x^n\,\rm dx $$ | $$ \frac{x^{n+1}}{n+1} + C \qquad n \ne -1 $$ |
| $$ \int \frac{1}{x}\,\rm dx $$ | $$ \ln \left| x \right| + C \qquad x \ne 0 $$ |
| $$ \int \frac{1}{x-a} \, \rm dx $$ | $$ \ln | x-a | + C \qquad x \ne a $$ |
| $$ \int \frac{1}{(x-a)^n} \, \rm dx $$ | $$-\frac{1}{(n-1)(x-a)^{n-1}} + C \qquad n \ne 1 , x \ne a $$ |
| $$ \int \frac{1}{1+x^2} \, \rm dx $$ | $$ \operatorname{arctan}(x) + C $$ |
| $$ \int \frac{1}{a^2+x^2} \, \rm dx $$ | $$ \frac{1}{a}\operatorname{arctan}{ \left( \frac{x}{a} \right) } + C \qquad a \ne 0 $$ |
| $$ \int \frac{1}{1-x^2} \, \rm dx $$ | $$ \frac{1}{2} \ln { \left| \frac{x+1}{x-1} \right| } + C $$ |
| $$ \int \ln (x)\,\rm dx $$ | $$ x \ln (x) - x + C $$ |
| $$ \int \log_b (x)\,\rm dx $$ | $$ x \log_b (x) - x \log_b (e) + C $$ |
| $$ \int e^x\,\rm dx $ | $$ 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 $$ |
The primitive calculation of some functions within dCode calculator can involve the functions denoted $ F $ and $ E $ respectively first and second kind of elliptic integrals, or $ Ci $ and $ Si $ respectively Cosine Integral and Sine Integral, or $ Li_2 $ the Spence's function or $ B_x $ the Euler Beta function.
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