Integral (antiderivative) calculator
In mathematical analysis, the integral is widely used - a continuous analogue of a sum, applicable for calculating areas, volumes, masses, distances and other non-constant (changeable) quantities.
For example, the speed of a vehicle, which can change many times while moving, or the frequency of a processor, which adapts to the computational processes being performed. It is impossible to describe these quantities as a fixed value, since they constantly change in the range from minimum to maximum, but this can be easily done using an integral.
Depending on whether the measured quantity has fixed limits, a definite and indefinite integral are distinguished. The first has them, but the second does not. The essence of integration remains the same.
In simple terms, this is a set of operations of multiplication of several terms with their subsequent summation, or the sum of an infinite number of multiplications performed with infinitesimal terms. Today integration is widely used for:
- Finding the areas of complex geometric figures for which it is impossible to derive a specific formula like S = a × b or S = π × r².
- Calculation of the masses of bodies with uneven density.
- Determination of distances traveled at varying speeds.
In mathematics (and other sciences), an integral is denoted by an elongated letter ∫, derived from the Latin S (summa). In essence, an integral is the sum of many multiplied terms. Moreover, ideal integration (without errors) can be carried out in relation to both finite and infinite quantities.
History of integral calculus
Although the very concept of “integral” did not yet exist, its principle began to be used back in Ancient Greece. Thus, Archimedes used to find the area of circles a method that was as close as possible to modern integration, namely the exhaustion method.
It consisted in fitting a sequence of other figures into a regular circle, followed by determining the limit of their areas. A direct analogy to these calculations is finding the limit of an infinite sum using integration.
Initially, the method was used only in geometry, but then found application in mechanics, economics, astronomy and other sciences. And its modern name, “integration,” arose only in the 17th century: during the research of European scientists Isaac Newton and Gottfried Wilhelm Leibniz. The integral began to be used in differential calculus systems, and it received a clear mathematical definition - “antiderivative of a function.”
In simple terms, an integral in geometry is the area of a curvilinear figure. The indefinite integral is the entire area, and the definite integral is the area in a given area. Accordingly, the process of finding the derivative is called differentiation, and finding the antiderivative is called integration.
Rules for integrating functions
When working with integrals, you can use transformation formulas, provided that they use the constant C. It is determined if the value of the integral at a specific (arbitrarily taken) point is known.
Since each function has an infinite number of antiderivatives, knowing the value of C, you can transform integral formulas in the following ways:
- ∫Сf(x)dx = C∫f(x)dx.
- ∫f(x) + g(x)dx = ∫f(x)dx + ∫g(x)dx.
- ∫f(x)g(x)dx = f(x)∫g(x)dx − ∫(∫g(x)dx)df(x).
- ∫f(ax + b)dx = (1/a)F(ax + b) + C.
Integrals of logarithmic and exponential functions can also be transformed:
- ∫lnxdx = xlnx − x + C.
- ∫(dx/xlnx) = ln|lnx| + C.
- ∫logₐxdx = xlogₐx − x logₐe + C = x((lnx−1)/lnb) + C.
- ∫eⁿdx = eⁿ + C.
- ∫aⁿdx = aⁿ/lna + C.
In trigonometry, at least 15 formulas for transforming integrals are used, the simplest of which are:
- ∫sinxdx = −cosx + C.
- ∫cosxdx = sinx + C.
- ∫tgxdx = −ln|cosx| + C.
Similar formulas exist for secants, cosecants, arctangents, and so on. Only a computer, or rather a special application with an integration function, can calculate them quickly (after substituting variables).
To quickly calculate a definite or indefinite integral, or an antiderivative of a function, use our calculator. It is enough to substitute numerical values into it and select calculation parameters. The result will be displayed on the screen in a split second, which will save you from the need to carry out long and complex calculations.
