Tool to find primitives of a function. Integration of a function is the calculation of all its primitives, the inverse of the derivative.
Primitives Functions - dCode
Tag(s) : Mathematics
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Primitives Functions
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Primitive Function Calculator
Tool to find primitives of a function. 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 \( F \), defined and differentiable over \( I \), which derivative is \( f \), ie. \( F'(x) = f(x) \).
Example: Consider \( f(x) = x^2+sin(x) \) the 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 talks about finding the function calculating the area under the curve.
What is the list of common primitives?
| Function | Primitive |
|---|---|
| $$ \int \,\rm dx$$ | $$x + C$$ |
| $$ \int x^n\,\rm dx$$ | $$ \frac{x^{n+1}}{n+1} + C \qquad n \ne -1 $$ |
| $$ \int \frac{1}{x}\,\rm dx$$ | $$\ln \left| x \right| + C \qquad x \ne 0 $$ |
| $$ \int \frac{1}{x-a} \, \rm dx $$ | $$\ln | x-a | + C \qquad x \ne a $$ |
| $$ \int \frac{1}{(x-a)^n} \, \rm dx$$ | $$-\frac{1}{(n-1)(x-a)^{n-1}} + C \qquad n \ne 1 , x \ne a $$ |
| $$ \int \frac{1}{1+x^2} \, \rm dx$$ | $$\operatorname{arctan}(x) + C $$ |
| $$ \int \frac{1}{a^2+x^2} \, \rm dx$$ | $$\frac{1}{a}\operatorname{arctan}{ \left( \frac{x}{a} \right) } + C \qquad a \ne 0 $$ |
| $$ \int \frac{1}{1-x^2} \, \rm dx$$ | $$\frac{1}{2} \ln { \left| \frac{x+1}{x-1} \right| } + C $$ |
| $$ \int \ln (x)\,\rm dx$$ | $$x \ln (x) - x + C $$ |
| $$ \int \log_b (x)\,\rm dx$$ | $$x \log_b (x) - x \log_b (e) + C $$ |
| $$ \int e^x\,\rm dx$$ | $$e^x + C $$ |
| $$ \int a^x\,\rm dx$$ | $$\frac{a^x}{\ln (a)} + C \qquad a > 0 , a \ne 1 $$ |
| $$ \int {1 \over \sqrt{1-x^2}} \, \rm dx$$ | $$\operatorname{arcsin} (x) + C $$ |
| $$ \int {-1 \over \sqrt{1-x^2}} \, \rm dx$$ | $$\operatorname{arccos} (x) + C $$ |
| $$ \int {x \over \sqrt{x^2-1}} \, \rm dx$$ | $$\sqrt{x^2-1} + C $$ |
| $$ \int \sin(x)\,\rm dx $$ | $$ -\cos(x)+C $$ |
| $$ \int \cos(x)\,\rm dx $$ | $$ \sin(x)+C $$ |
| $$ \int \tan(x)\,\rm dx $$ | $$ -\ln|\cos(x)|+C $$ |
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