Integral
lastmod 2018-08-27created 2014-09-0314 edits
By the classical definition in Riemann integration, the integral is the limit of the Riemann sums of the curve formed by the range corresponding to an interval [a, b], a subset of the domain of a continuous function f(x) that can be represented in a measure space using the real-number scale.
The definite integral of a function \(f(x)\) continuous on the closed interval \([a,b]\) can be written as follows.
\[ \int_{a}^{b} f(x)dx=\lim_{n \to \infty} \sum_{k=1}^{n}f(x_k) \Delta x \quad \left( x_{k} = a + k \Delta x,\ \Delta x =\frac {b-a} {n} \right) \]
Table of integrals
Various integration results and formulas are summarized below.
- A definite integral is expressed by first finding the indefinite integral and then taking the difference of the values at the upper and lower limits, as follows.
- \(\displaystyle \int^{b}_a f\left(x\right) dx = F\left(b\right) - F\left(a\right) , \frac {d}{ dx}F\left(x\right) = f\left(x\right)\)
Antiderivative
In calculus, an antiderivative (逆導函數; English: antiderivative), primitive function (原函數), or indefinite integral of a function is a function whose derivative is that function. Indefinite integration (不定積分; English: indefinite integral) is the operation of finding (all) antiderivatives. If a function f(x) is the derivative of F(x), then F(x) is an antiderivative of f(x). However, it is not the only antiderivative; adding an arbitrary constant to it gives all the antiderivatives. This is written as follows.
\[ \int f(x)dx = F(x) + C \]
For this reason, indefinite integration is also called the inverse operation of differentiation.
Examples from differentiation
From examples of differentiation, we can obtain examples of indefinite integration, its inverse operation. For example, the derivative of \(f(x) = x\) is \(f'(x) = 1\), so the following holds.
\[ \int dx = x +C \]
For the monomial \(f(x) = x^r\), the following holds.
\[ \int x^r\, dx = \frac{1}{r+1} x^{r+1} + C , (\ r\ne -1) \]
In practice, verifying an indefinite integral through differentiation is easy, but it is not a general method of solution.
Worked examples
- [Recommended] Definite integrals and infinite series 1
Notation and computation
The notation \(\int\) commonly used for integrals today was proposed by Leibniz. Given an integrable function f(x), the integral is written as follows.
\[ \int_{a}^{b} f(x) dx \] (the integral of the function \(f(x)=x\) over the interval \([a, b]\))
By the fundamental theorem of calculus, integration is the inverse of differentiation. That is, there exists a primitive function \(F(x)\) whose derivative is the function \(f(x)=x\). The process of finding the primitive function is called indefinite integration (不定積分). Expressed as an equation, the following relation holds.
\[ \frac{d}{dx} F(x) = f(x) \]
For example, if \(f(x)=x\), the primitive function \(F(x)\) can be found by reasoning as follows.
\[ \frac{d}{dx}F(x)=x \ \]
Here, since the primitive function \(F(x)\) whose derivative is \(f(x)=x\) is a function defined by a general polynomial,
\[ F(x)=A x^2 + B x + C \ \] we can see it takes this form. Therefore, using the rule for differentiating polynomials, \(2A=1, B=0\), and \(C\) is an arbitrary constant. That is,
\[ F(x)=\frac{1}{2} x^2 + C \ \] The arbitrary constant \(C\) appearing in the equation above is called the constant of integration. The constant of integration in the primitive function \(F(x)\) becomes \(0\) and vanishes when differentiated into the function \(f(x)\).2
In general,
\[ f(x)=a_0 x^n + a_1 x^{n-1} + \cdots + a_n \] (where \(n \ne -1\)) the primitive function F(x) of a function defined by a polynomial of this form takes the following form.
\[ F(x) = \frac{a_0}{n+1} x^{n+1} + \frac{a_1}{n} x^n + \cdots + a_n x + C \] (C is the constant of integration)
See also
Favorite site
- Wikipedia (en) description of the integral
- Wikipedia (ko) description of the integral 3
- Wikipedia (ko) Table of integrals
- [Recommended] Dark Programmer: Understanding differentiation and integration properly 4