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Definite Integral

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.

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Definite Integral -

Tag(s) : Functions, Symbolic Computation

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# Definite Integral

## Multiple Integral Calculator

### 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?

FunctionPrimitive
$$\int \,\rm dx$$$$x + C$$
$$\int x^n\,\rm dx$$$$\frac{x^{n+1}}{n+1} + C \qquad n \ne -1$$
$$\int \frac{1}{x}\,\rm dx$$$$\ln \left| x \right| + C \qquad x \ne 0$$
$$\int \frac{1}{x-a} \, \rm dx$$$$\ln | x-a | + C \qquad x \ne a$$
$$\int \frac{1}{(x-a)^n} \, \rm dx$$$$-\frac{1}{(n-1)(x-a)^{n-1}} + C \qquad n \ne 1 , x \ne a$$
$$\int \frac{1}{1+x^2} \, \rm dx$$$$\operatorname{arctan}(x) + C$$
$$\int \frac{1}{a^2+x^2} \, \rm dx$$$$\frac{1}{a}\operatorname{arctan}{ \left( \frac{x}{a} \right) } + C \qquad a \ne 0$$
$$\int \frac{1}{1-x^2} \, \rm dx$$$$\frac{1}{2} \ln { \left| \frac{x+1}{x-1} \right| } + C$$
$$\int \ln (x)\,\rm dx$$$$x \ln (x) - x + C$$
$$\int \log_b (x)\,\rm dx$$$$x \log_b (x) - x \log_b (e) + C$$
$$\int e^x\,\rm dx$$$$e^x + C$$
$$\int a^x\,\rm dx$$$$\frac{a^x}{\ln (a)} + C \qquad a > 0 , a \ne 1$$
$$\int {1 \over \sqrt{1-x^2}} \, \rm dx$$$$\operatorname{arcsin} (x) + C$$
$$\int {-1 \over \sqrt{1-x^2}} \, \rm dx$$$$\operatorname{arccos} (x) + C$$
$$\int {x \over \sqrt{x^2-1}} \, \rm dx$$$$\sqrt{x^2-1} + C$$
$$\int \sin(x)\,\rm dx$$$$-\cos(x)+C$$
$$\int \cos(x)\,\rm dx$$$$\sin(x)+C$$
$$\int \tan(x)\,\rm dx$$$$-\ln|\cos(x)|+C$$

Reminder: the derivative of the primitive of a function is the function itself.

### What are the functions E, F, I0, K0?

Calculation of some forms of integrals involve special functions such as $E$ and $F$ which are elliptic integrals or $I_0, I_n, J_0, J_n, K_0, K_n$ which are Bessel functions.

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