Definite Integral

The integral is the operator of the integration calculation in mathematics, for a function $ f(x) $ on a given interval $ [a, b] $, the integral is denoted $ \int_{a}^{b} f(x) , dx $

Integration is generally presented as a method of calculating the area under the curve of a function, but it can also be applied to the calculation of surfaces and volumes of solids.

Integral calculus is usually set to an interval and uses function primitives.

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 $$ \int^b_a f(x) \mathrm{ dx} = F(b)-F(a) $$

Enter the function, its lower and upper bounds and the variable to integrate, dCode will make the computation automatically.

The fundamental properties of integrals are:

— Linearity: $ \int_{a}^{b} (λf + μg)dx = λ\int_{a}^{b} f(x)dx + μ\int_{a}^{b} g(x)dx $

— Chasles relation: $ \int_{a}^{b} f(x)dx + \int_{b}^{c} f(x)dx = \int_{a}^{c} f(x)dx $

— Positivity: If f(x) ≥ 0 then $ \int_{a}^{b} f(x)dx ≥ 0 $

The complete list of common primitives is available on the function primitives page.

The integration involves the primitives of functions to perform the calculation. Primitives are a tool for calculating integrals.

The integral is the operator of the integration calculation, the derivative is the result of the differential calculation. Integral calculus and differential calculus are the 2 fields of infinitesimal calculus.

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.