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Informaţii, definiţii, teoreme, formule, exerciţii şi probleme rezolvate din matematica de liceu.

Data publicarii: 02 Aprilie, 2011

TEORIE

Definitie:

O functie f, definita pe intervalul I si cu valori in R, este primitivabila pe I,

daca există o functie F definita pe I si cu valori in R, derivabilă pe I şi F'(x) = f(x),

oricare ar fi xЄI; funcţia F se numeşte o primitivă a funcţiei f şi, evident, în acest caz,

există o  infinitate de primitive ale funcţiei f, mulţime care se numeşte

integrala nedefinită a funcţiei f; notatie: 

\int{f(x)}{dx}=\{F|F:{I}\rightarrow{R}\}.\int{f(x)}{dx}=\{F|F:{I}\rightarrow{R}\}.

Daca functia f:I - > R admite o primitiva F, atunci

\int{f(x)}{dx}=F+\mathcal{C},\int{f(x)}{dx}=F+\mathcal{C},

unde C = {f|f:I - > R, f(x) = cЄR, xЄI}, este

mulţimea tuturor funcţiilor constante definite pe I.

Primitive uzuale:

1)\;\int{x}^{n}{dx} =\frac{{x}^{n+1}}{n+1} +\mathcal{C},{x}\in{\mathbb{R}},\forall{n}\in{\mathbb{N}}1)\;\int{x}^{n}{dx} =\frac{{x}^{n+1}}{n+1} +\mathcal{C},{x}\in{\mathbb{R}},\forall{n}\in{\mathbb{N}} \Rightarrow \int{1}\cdot{dx} = x +\mathcal{C}, {x}\in{\mathbb{R}}.\Rightarrow \int{1}\cdot{dx} = x +\mathcal{C}, {x}\in{\mathbb{R}}.

2)\;\int{x}^{\alpha}{dx} = \frac{{x}^{\alpha+1}}{\alpha+1} +\mathcal{C}2)\;\int{x}^{\alpha}{dx} = \frac{{x}^{\alpha+1}}{\alpha+1} +\mathcal{C} , {x}\in{I\subset(o,\infty)},\forall{\alpha\in{\mathbb{R}}}\setminus{\begin{Bmatrix}-1\end{Bmatrix}}.{x}\in{I\subset(o,\infty)},\forall{\alpha\in{\mathbb{R}}}\setminus{\begin{Bmatrix}-1\end{Bmatrix}}.

3)\;\int\frac{1}{x}{dx}=\ln{|x|}+\mathcal{C}3)\;\int\frac{1}{x}{dx}=\ln{|x|}+\mathcal{C} , {x}\in{I\subset(0,\infty)},{ou}\,{x}\in{I\subset(-\infty,0)}.{x}\in{I\subset(0,\infty)},{ou}\,{x}\in{I\subset(-\infty,0)}.

4)\; \int{a}^{x}{dx}=\frac{{a}^{x}}{\ln{a}}+\mathcal{C},4)\; \int{a}^{x}{dx}=\frac{{a}^{x}}{\ln{a}}+\mathcal{C}, {x}\in{\mathbb{R}},{a>0},a\neq{1}\Rightarrow \int{e}^{x}{dx}={e}^{x}+\mathcal{C},{x}\in{\mathbb{R}}.{x}\in{\mathbb{R}},{a>0},a\neq{1}\Rightarrow \int{e}^{x}{dx}={e}^{x}+\mathcal{C},{x}\in{\mathbb{R}}.

5)\;\int\frac{1}{{x}^{2}-{a}^{2}}{dx}=\frac{1}{2a}\cdot\ln{|\frac{x-a}{x+a}|}+\mathcal{C},5)\;\int\frac{1}{{x}^{2}-{a}^{2}}{dx}=\frac{1}{2a}\cdot\ln{|\frac{x-a}{x+a}|}+\mathcal{C}, {x}\in{I=(-\infty,-a)}\;ou\;{x}\in{I=(-\infty,-a)}\;ou\; {x}\in{I=(-a,a)}\;ou\;{x}\in{I=(-a,a)}\;ou\; {x}\in{I=(a,\infty)},{a}\in{(0,\infty)}.{x}\in{I=(a,\infty)},{a}\in{(0,\infty)}.

6)\;\int\frac{1}{{x}^{2}+{a}^{2}}{dx} =\frac{1}{a}\cdot{arctg}{\frac{x}{a}}+\mathcal{C},{x}\in{\mathbb{R}}, {a}\neq{0}.6)\;\int\frac{1}{{x}^{2}+{a}^{2}}{dx} =\frac{1}{a}\cdot{arctg}{\frac{x}{a}}+\mathcal{C},{x}\in{\mathbb{R}}, {a}\neq{0}.

7)\;\int\frac{1}{\sqrt{{a}^{2}-{x}^{2}}}{dx}=\arcsin{\frac{x}{a}}+\mathcal{C},{x}\in{I\subset(-a,a)},{a}\neq{0}.7)\;\int\frac{1}{\sqrt{{a}^{2}-{x}^{2}}}{dx}=\arcsin{\frac{x}{a}}+\mathcal{C},{x}\in{I\subset(-a,a)},{a}\neq{0}.

8)\;\int\frac{1}{\sqrt{{x}^{2}+{a}^{2}}}{dx}=\ln{(x+\sqrt{{x}^{2}+{a}^{2}})}+\mathcal{C},{x}\in{\mathbb{R}}, {a}\neq{0}.8)\;\int\frac{1}{\sqrt{{x}^{2}+{a}^{2}}}{dx}=\ln{(x+\sqrt{{x}^{2}+{a}^{2}})}+\mathcal{C},{x}\in{\mathbb{R}}, {a}\neq{0}.

9)\;\int\frac{1}{\sqrt{{x}^{2}-{a}^{2}}}{dx}=\ln{|x+\sqrt{{x}^{2}-{a}^{2}}|}+\mathcal{C},{x}\in{I}\subset(-\infty,-a)\;ou\;9)\;\int\frac{1}{\sqrt{{x}^{2}-{a}^{2}}}{dx}=\ln{|x+\sqrt{{x}^{2}-{a}^{2}}|}+\mathcal{C},{x}\in{I}\subset(-\infty,-a)\;ou\; {x}\in{I}\subset(a,\infty),{a}>0.{x}\in{I}\subset(a,\infty),{a}>0.

10)\;\int\sin{x}{dx}= -\cos{x}+\mathcal{C},{x}\in{\mathbb{R}}.10)\;\int\sin{x}{dx}= -\cos{x}+\mathcal{C},{x}\in{\mathbb{R}}.

11)\;\int\cos{x}{dx}=\sin{x}+\mathcal{C},{x}\in{\mathbb{R}}.11)\;\int\cos{x}{dx}=\sin{x}+\mathcal{C},{x}\in{\mathbb{R}}.

12)\;\int\frac{1}{{\cos}^{2}{x}}{dx}={tgx}+\mathcal{C},{x}\in{I\subset((2k+1)\frac{\pi}{2},(2k+3)\frac{\pi}{2})}, {k}\in{\mathbb{Z}}.12)\;\int\frac{1}{{\cos}^{2}{x}}{dx}={tgx}+\mathcal{C},{x}\in{I\subset((2k+1)\frac{\pi}{2},(2k+3)\frac{\pi}{2})}, {k}\in{\mathbb{Z}}.

13)\;\int\frac{1}{{\sin}^{2}{x}}{dx}= -{ctgx}+\mathcal{C},{x}\in{I\subset(k\pi,(k+1)\pi)}, {k}\in{\mathbb{Z}}.13)\;\int\frac{1}{{\sin}^{2}{x}}{dx}= -{ctgx}+\mathcal{C},{x}\in{I\subset(k\pi,(k+1)\pi)}, {k}\in{\mathbb{Z}}.

14)\;\int{tgx}{dx}= -\ln{|\cos{x}|}+\mathcal{C},{x}\in{I\subset((2k+1)\frac{\pi}{2},(2k+3)\frac{\pi}{2})}, {k}\in{\mathbb{Z}}14)\;\int{tgx}{dx}= -\ln{|\cos{x}|}+\mathcal{C},{x}\in{I\subset((2k+1)\frac{\pi}{2},(2k+3)\frac{\pi}{2})}, {k}\in{\mathbb{Z}}

15)\;\int{ctgx}{dx} = \ln{|\sin{x}|}+\mathcal{C},{x}\in{I\subset(k\pi,(k+1)\pi)},{k}\in{\mathbb{Z}}.15)\;\int{ctgx}{dx} = \ln{|\sin{x}|}+\mathcal{C},{x}\in{I\subset(k\pi,(k+1)\pi)},{k}\in{\mathbb{Z}}.

16)\;\int\sinh{x}{dx}=\cosh{x}+\mathcal{C},{x}\in{\mathbb{R}},\;{ou}\;\sinh{x}=\frac{{e}^{x}-{e}^{-x}}{2}.16)\;\int\sinh{x}{dx}=\cosh{x}+\mathcal{C},{x}\in{\mathbb{R}},\;{ou}\;\sinh{x}=\frac{{e}^{x}-{e}^{-x}}{2}.

17)\;\int\cosh{x}{dx}=\sinh{x}+\mathcal{C},{x}\in{\mathbb{R}},\;{ou}\;\cosh{x}= \frac{{e}^{x}+{e}^{-x}}{2}.17)\;\int\cosh{x}{dx}=\sinh{x}+\mathcal{C},{x}\in{\mathbb{R}},\;{ou}\;\cosh{x}= \frac{{e}^{x}+{e}^{-x}}{2}.

Observatii:

  • O functie f:I - > R, care admite primitive pe I, are proprietatea lui Darboux pe I. 
  • Rezulta ca daca f:I - > R nu are proprietatea lui Darboux pe I, atunci f nu admite primitive pe I.
  • Orice functie continua, f:I - > R, admite primitive pe I.

Metoda integrarii prin parti:

\int{f(x)g\int{f(x)g'(x)}{dx}={f(x)g(x)}-\int{f'(x)g(x)}{dx}.

Prima metoda a schimbarii de variabila:

Fie functiile φ:I - > J, h:J - > R, unde I si J sunt intervale din R.

Daca: 

a) φ este derivabila si

b) h admite o primitiva H, atunci functia

1) (h o φ)·φ' admite primitive pe I si

2)\;\int{h(\varphi(x))\varphi2)\;\int{h(\varphi(x))\varphi'(x)}{dx}=H(\varphi(x))+\mathcal{C}.

Primitive uzuale: 

1)\;\int{{\varphi}^{n}}(x){{\varphi}^{1)\;\int{{\varphi}^{n}}(x){{\varphi}^{'}}(x){dx}=\frac{{{\varphi}^{n+1}}(x)}{n+1}+\mathcal{C},{n}\in{\mathbb{N}}.

2)\;\int{{\varphi}^{\alpha}}(x){{\varphi}^{2)\;\int{{\varphi}^{\alpha}}(x){{\varphi}^{'}}(x){dx}=\frac{{{\varphi}^{\alpha+1}}(x)}{\alpha+1}+\mathcal{C},{\alpha}\neq{-1},\varphi(I)\subset(0,\infty).

3)\;\int{\frac{{\varphi}^{3)\;\int{\frac{{\varphi}^{'}(x)}{{\varphi}{(x)}}}{dx}= \ln{|{\varphi}(x)|}+\mathcal{C},{\varphi}(x)\neq{0}, {x}\in{I}.

4)\;\int{a}^{{\varphi}(x)}{{\varphi}^{4)\;\int{a}^{{\varphi}(x)}{{\varphi}^{'}}(x){dx}=\frac{{a}^{{\varphi}(x)}}{\ln{a}}+\mathcal{C},a > 0,{a}\neq{1}.

5)\;\int{\frac{{{\varphi}^{5)\;\int{\frac{{{\varphi}^{'}}{(x)}}{{{\varphi}^{2}}{(x)}-{a}^{2}}}{dx}=\frac{1}{2a} \ln{|\frac{{\varphi}{(x)}-a}{{\varphi}{(x)}+a}|}+\mathcal{C},{\varphi}{(x)}\neq{\pm{a}},{x}\in{I},{a}\neq{0}.\ln{|\frac{{\varphi}{(x)}-a}{{\varphi}{(x)}+a}|}+\mathcal{C},{\varphi}{(x)}\neq{\pm{a}},{x}\in{I},{a}\neq{0}.

6)\;\int{\frac{{{\varphi}^{6)\;\int{\frac{{{\varphi}^{'}}{(x)}}{{{\varphi}^{2}}{(x)}+{a}^{2}}}{dx}=\frac{1}{a} {arctg}{\frac{{\varphi}(x)}{a}}+\mathcal{C}, {a}\neq{0}.{arctg}{\frac{{\varphi}(x)}{a}}+\mathcal{C}, {a}\neq{0}.

7)\;\int{\frac{{{\varphi}^{7)\;\int{\frac{{{\varphi}^{'}}{(x)}}{\sqrt{{a}^{2}-{{\varphi}^{2}}{(x)}}}}{dx}=\arcsin{\frac{{\varphi}{(x)}}{a}} + \mathcal{C}, a > 0, \varphi{(I)}\subset(-a,a).

8)\;\int{\frac{{{\varphi}^{8)\;\int{\frac{{{\varphi}^{'}}{(x)}}{\sqrt{{{\varphi}^{2}}{(x)}+{a}^{2}}}}{dx}=\ln{[{\varphi}{(x)}+\sqrt{{{\varphi}^{2}}{(x)}+{a}^{2}}}]+\mathcal{C},{a}\neq{0}.

9)\;\int{\frac{{{\varphi}^{9)\;\int{\frac{{{\varphi}^{'}}{(x)}}{\sqrt{{{\varphi}^{2}}{(x)}-{a}^{2}}}}{dx}=\ln{|{\varphi}{(x)}+\sqrt{{{\varphi}^{2}}{(x)}-{a}^{2}}}|+\mathcal{C},a >0, \varphi(I)\subset(-\infty,-a)\;ou\;\varphi(I)\subset(a,\infty).\varphi(I)\subset(-\infty,-a)\;ou\;\varphi(I)\subset(a,\infty).

10)\;\int\sin{\varphi(x)}{{\varphi}^{10)\;\int\sin{\varphi(x)}{{\varphi}^{'}}(x){dx}= -\cos{\varphi(x)}+\mathcal{C}.

11)\;\int\cos{\varphi(x)}{{\varphi}^{11)\;\int\cos{\varphi(x)}{{\varphi}^{'}}(x){dx}=\sin{\varphi(x)}+\mathcal{C}.

12)\;\int{\frac{{\varphi}^{12)\;\int{\frac{{\varphi}^{'}(x)}{{{\cos}^{2}}\varphi{(x)}}}{dx}={tg}{\varphi{(x)}}+\mathcal{C}, \varphi{(x)}\neq{(2k+1)}\cdot{\frac{\pi}{2}}, {k}\in{\mathbb{Z}},{x}\in{I}.\varphi{(x)}\neq{(2k+1)}\cdot{\frac{\pi}{2}}, {k}\in{\mathbb{Z}},{x}\in{I}.

13)\;\int{\frac{{\varphi}^{13)\;\int{\frac{{\varphi}^{'}(x)}{{{\sin}^{2}}\varphi{(x)}}}{dx}= -{ctg}{\varphi{(x)}}+\mathcal{C}, \varphi{(x)}\neq{k\pi}, {k}\in{\mathbb{Z}}, {x}\in{I}.\varphi{(x)}\neq{k\pi}, {k}\in{\mathbb{Z}}, {x}\in{I}.

14)\;\int{tg}{\varphi{(x)}}{{\varphi}^{14)\;\int{tg}{\varphi{(x)}}{{\varphi}^{'}}{(x)}{dx}= -\ln{|\cos{\varphi{(x)}}|}\mathcal{C}, \varphi{(x)}\neq{(2k+1)\cdot{\frac{\pi}{2}}},{k}\in{\mathbb{Z}},{x}\in{I}.\varphi{(x)}\neq{(2k+1)\cdot{\frac{\pi}{2}}},{k}\in{\mathbb{Z}},{x}\in{I}.

15)\;\int{ctg}{\varphi{(x)}}{{\varphi}^{15)\;\int{ctg}{\varphi{(x)}}{{\varphi}^{'}}{(x)}{dx}=\ln{|\sin{\varphi{(x)}}|}+\mathcal{C}, \varphi{(x)}\neq{k\pi},{k}\in{\mathbb{Z}},{x}\in{I}.\varphi{(x)}\neq{k\pi},{k}\in{\mathbb{Z}},{x}\in{I}.

16)\;\int\sinh{\varphi(x)}{\varphi}^{16)\;\int\sinh{\varphi(x)}{\varphi}^{'}{(x)}{dx}=\cosh{\varphi(x)}+\mathcal{C}.

17)\;\int\cosh{\varphi(x)}{\varphi}^{17)\;\int\cosh{\varphi(x)}{\varphi}^{'}{(x)}{dx}=\sinh{\varphi(x)}+\mathcal{C}.

A doua metoda a schimbarii de variabila:

Fie I si J intervale in R si functiile φ:I - > J, h:J - > R.

Daca:

a) Functia φ este bijectiva, derivabila, astfel incat derivata φ'(t) este nenula,

oricare ar fi tЄI, si

b) Functia h = (f o φ)·φ' admite primitive (fie H o primitiva a sa), atunci:

1) Functia f admite primitive,

2) Functia {H}\circ{\varphi}^{-1}{H}\circ{\varphi}^{-1} este o primitiva a functiei f, adica:

\int{f(x)}{dx}=({H}\circ{\varphi}^{-1})(x)+\mathcal{C}.\int{f(x)}{dx}=({H}\circ{\varphi}^{-1})(x)+\mathcal{C}.  

Postat în: PRIMITIVE-liceu

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