In general, a function F ( x ) that satisfies F '( x ) = f ( x ) for a function f ( x ) is called a primitive function of f ( x ), and finding a primitive function of a given function is called integration. This is the inverse operation of differentiation. For example, in , x 3 is a primitive function of 3 x 2. A primitive function does not necessarily exist for any function f ( x ), but if it is a continuous function, then a primitive function exists. If F ( x ) is a primitive function of f ( x ), then all other primitive functions are given by F ( x ) + C ( C is an arbitrary constant). In general, all primitive functions of f ( x ) are collectively represented by the symbol ∫ f ( x ) dx . For continuous functions, a primitive function is the same as an indefinite integral (Fundamental Theorem of Calculus), so the term primitive function is not used and it is often called an indefinite integral or simply an integral, but in theory the two are different concepts. Source: Encyclopaedia Britannica Concise Encyclopedia About Encyclopaedia Britannica Concise Encyclopedia Information |
一般に関数 f(x) に対し,F'(x)=f(x) を満たす関数 F(x) を,f(x) の原始関数といい,与えられた関数の原始関数を求めることを積分するという。これは微分の逆演算である。たとえば において,x3 は 3x2 の原始関数である。任意の関数 f(x) に対しては,必ずしも原始関数が存在するとはかぎらないが,連続関数ならば,その原始関数が存在する。 F(x) が f(x) の1つの原始関数ならば,他の原始関数はすべて F(x)+C ( C は任意の定数) で与えられる。一般に f(x) の原始関数を一括して,記号 ∫f(x)dx で表わす。連続関数については,原始関数は不定積分と一致する (微分積分学の基本定理) ので,原始関数という用語を使わず,不定積分あるいは単に積分と呼ぶことが多いが,理論上両者は別の概念である。
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