Proprietăţi ale integralelor nedefinite
modificare
a) Dacă f : I → R {\displaystyle f:I\to \mathbb {R} } admite primitive pe I , iar a ∈ R , {\displaystyle a\in \mathbb {R} ,} atunci şi a f {\displaystyle \mathrm {a} f} admite primitive şi:∫ a f ( x ) d x = a ∫ f ( x ) d x . {\displaystyle \int af(x)\,\mathrm {d} x=a\int f(x)\,\mathrm {d} x.}
b) Dacă f şi g admit primitive pe I atunci f + g {\displaystyle f+g} admite primitive pe I şi:∫ ( f ( x ) + g ( x ) ) d x = ∫ f ( x ) d x + ∫ g ( x ) d x {\displaystyle \int \left(f(x)+g(x)\right)\,\mathrm {d} x=\int f(x)\mathrm {d} x+\int g(x)\mathrm {d} x}
Integrale nedefinite utilizate frecvent
modificare
f : R → R , f ( x ) = x n , n ∈ N {\displaystyle f:\mathbb {R} \to \mathbb {R} ,\;f(x)=x^{n},\;n\in \mathbb {N} \;\;}
∫ x n d x = x n + 1 n + 1 + C . {\displaystyle \int x^{n}\mathrm {d} x={\frac {x^{n+1}}{n+1}}+{\mathcal {C}}.}
f : I → R , I ⊂ ( 0 , ∞ ) , f ( x ) = x a , a ∈ R ∖ { 1 } {\displaystyle f:I\to \mathbb {R} ,\;I\subset (0,\infty )\;,\;f(x)=x^{a}\;,\;a\in \mathbb {R} \setminus \{1\}}
∫ x a d x = x a + 1 a + 1 + C . {\displaystyle \int x^{a}\mathrm {d} x={\frac {x^{a+1}}{a+1}}+{\mathcal {C}}.}
f : I → R , I ⊂ R ∗ , f ( x ) = 1 x {\displaystyle f:I\to \mathbb {R} \;,\;I\subset \mathbb {R} ^{*}\;,\;f(x)={\frac {1}{x}}}
∫ 1 x d x = ln | x | + C . {\displaystyle \int {\frac {1}{x}}\mathrm {d} x=\ln |x|+{\mathcal {C}}.}
f : R → R , f ( x ) a x , a > 0 , a ≠ 1 {\displaystyle f:\mathbb {R} \to \mathbb {R} \;,\;f(x)a^{x}\;,\;a>0\;,\;a\neq 1}
∫ a x d x = a x ln a + C . {\displaystyle \int a^{x}\mathrm {d} x={\frac {a^{x}}{\ln a}}+{\mathcal {C}}.}
f : I → R , I ⊂ R ∖ { − a , a } , f ( x ) = 1 x 2 − a 2 , a ≠ 0 {\displaystyle f:I\to \mathbb {R} \;,\;I\subset \mathbb {R} \setminus \{-a,a\}\,,\,f(x)={\frac {1}{x^{2}-a^{2}}}\,,\,a\neq 0}
∫ d x x 2 − a 2 = 1 2 a ln | x − a x + a | + C . {\displaystyle \int {\frac {\mathrm {d} x}{x^{2}-a^{2}}}={\frac {1}{2a}}\ln \left|{\frac {x-a}{x+a}}\right|+{\mathcal {C}}.}
{\displaystyle }
{\displaystyle }
{\displaystyle }
{\displaystyle }