Tool to calculate the integral of a function. The computation of an definite integral over an interval consist in measuring the area under the curve of the function to integrate.
Definite Integral - dCode
Tag(s) : Functions, Symbolic Computation
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Definite Integral
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Primitive Calculator
This page deals with integral calculation on an interval. For the general case, with infinite bounds, see the calculation of primitives.
Definite Integral Calculator
Tool to calculate the integral of a function. The computation of an definite integral over an interval consist in measuring the area under the curve of the function to integrate.
Answers to Questions
How to calculate a definite integral over an interval?
To perform an integral calculation (integration), first, calculate the corresponding primitive function.
For a function \( f(x) \) to be integrated over \( [a;b] \) and \( F(x) \) the primitive of \( f(x) \). Then $$ \int^b_a f(x) \mathrm{ dx} = F(b)-F(a) $$
Example: Integrate \( f(x) = x \) over the interval \( [0;1] \). Calculate its primitive \( F(x) = \frac{1}{2} x^2 \) and so integral $$ \int^1_0 f(x) \mathrm {dx} = F(1) - F(0) = \frac{1}{2} $$
Enter the function, its lower and upper bounds and the variable to integrate, dCode will make the computation.
What is the list of common primitives?
| Function | Primitive |
|---|---|
| $$ \int \,\rm dx$$ | $$x + C$$ |
| $$ \int x^n\,\rm dx$$ | $$ \frac{x^{n+1}}{n+1} + C \qquad n \ne -1 $$ |
| $$ \int \frac{1}{x}\,\rm dx$$ | $$\ln \left| x \right| + C \qquad x \ne 0 $$ |
| $$ \int \frac{1}{x-a} \, \rm dx $$ | $$\ln | x-a | + C \qquad x \ne a $$ |
| $$ \int \frac{1}{(x-a)^n} \, \rm dx$$ | $$-\frac{1}{(n-1)(x-a)^{n-1}} + C \qquad n \ne 1 , x \ne a $$ |
| $$ \int \frac{1}{1+x^2} \, \rm dx$$ | $$\operatorname{arctan}(x) + C $$ |
| $$ \int \frac{1}{a^2+x^2} \, \rm dx$$ | $$\frac{1}{a}\operatorname{arctan}{ \left( \frac{x}{a} \right) } + C \qquad a \ne 0 $$ |
| $$ \int \frac{1}{1-x^2} \, \rm dx$$ | $$\frac{1}{2} \ln { \left| \frac{x+1}{x-1} \right| } + C $$ |
| $$ \int \ln (x)\,\rm dx$$ | $$x \ln (x) - x + C $$ |
| $$ \int \log_b (x)\,\rm dx$$ | $$x \log_b (x) - x \log_b (e) + C $$ |
| $$ \int e^x\,\rm dx$$ | $$e^x + C $$ |
| $$ \int a^x\,\rm dx$$ | $$\frac{a^x}{\ln (a)} + C \qquad a > 0 , a \ne 1 $$ |
| $$ \int {1 \over \sqrt{1-x^2}} \, \rm dx$$ | $$\operatorname{arcsin} (x) + C $$ |
| $$ \int {-1 \over \sqrt{1-x^2}} \, \rm dx$$ | $$\operatorname{arccos} (x) + C $$ |
| $$ \int {x \over \sqrt{x^2-1}} \, \rm dx$$ | $$\sqrt{x^2-1} + C $$ |
| $$ \int \sin(x)\,\rm dx $$ | $$ -\cos(x)+C $$ |
| $$ \int \cos(x)\,\rm dx $$ | $$ \sin(x)+C $$ |
| $$ \int \tan(x)\,\rm dx $$ | $$ -\ln|\cos(x)|+C $$ |
Reminder: the derivative of the primitive of a function is the function itself.
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