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»INTEGRALE DEFINITE

Data publicarii: 15 Iulie, 2010

EXEMPLUL 1

Suport teoretic:

Integrale definite, metoda intai a schimbarii de variabila, metoda integrarii prin parti.

Enunt:

Sa se calculeze integrala definita:

$I=\int_1^{e^{\frac{\pi}{2}}}{{sin}(lnx)}{dx}.$ I=\int_1^{e^{\frac{\pi}{2}}}{{sin}(lnx)}{dx}.

Raspuns:

$I=\frac{\sqrt{e^{\pi}+1}+1}{2}.$ I=\frac{\sqrt{e^{\pi}+1}+1}{2}.

Rezolvare:

Folosind prima metoda a schimbarii de variabila la integrala definita, avem succesiv:

${{lnx}=t}\Leftrightarrow{x=e^t},$ {{lnx}=t}\Leftrightarrow{x=e^t},

de unde, pentru x = 1 rezulta t = 0 si pentru

$x=e^{\frac{\pi}{2}},$ x=e^{\frac{\pi}{2}},

rezulta t = π/2, deci:

$I=\int_0^{\frac{\pi}{2}}{e^t}{sint}{dt}=\int_0^{\frac{\pi}{2}}({e^t})^{$ I=\int_0^{\frac{\pi}{2}}{e^t}{sint}{dt}=\int_0^{\frac{\pi}{2}}({e^t})^{'}{sint}{dt}=({e^t}{sint})|_0^{\frac{\pi}{2}}-\int_0^{\frac{\pi}{2}}{e^t}{cost}{dt},

(conform metodei de integrare prin parti la integrala definita). Rezulta apoi:

$I=e^{\frac{\pi}{2}}-\int_0^{\frac{\pi}{2}}{e^t}{cost}{dt}=e^{\frac{\pi}{2}}-J,\;unde\;J=\int_0^{\frac{\pi}{2}}{e^t}{cost}{dt},\;deci\;I+J=e^{\frac{\pi}{2}}.$ I=e^{\frac{\pi}{2}}-\int_0^{\frac{\pi}{2}}{e^t}{cost}{dt}=e^{\frac{\pi}{2}}-J,\;unde\;J=\int_0^{\frac{\pi}{2}}{e^t}{cost}{dt},\;deci\;I+J=e^{\frac{\pi}{2}}.

Se calculeaza in continuare I + J, in ideea obtinerii unui sistem de ecuatii cu

necunoscutele I si J si obtinem:

$I-J=\int_0^{\frac{\pi}{2}}{e^t}({sint}-{cost}){dt}=\cdots=e^{\frac{\pi}{2}}+1-I-J,$ I-J=\int_0^{\frac{\pi}{2}}{e^t}({sint}-{cost}){dt}=\cdots=e^{\frac{\pi}{2}}+1-I-J,