Question

Cum se rezolva aceasta integrala?

Answer 1

Răspuns:

Explicație pas cu pas:

$\displaystyle\texttt{In cazul in care avem de rezolvat o integrala cu capetele egale}\\\texttt{avem de verificat doua lucruri :}\\\texttt{Daca functia este }\bold{impara}\texttt{ atunci }\int_{-a}^af(x)dx=0\\\texttt{Daca functia este }\bold{para}\texttt{ atunci }\int_{-a}^af(x)dx=2\int_0^af(x)dx\\\texttt{In cazul in care niciunul dintre aceste cazuri nu sunt}\\\texttt{satisfacute vom face schimbarea de variabila }x=-t.\\\texttt{In cazul de fata niciuna dintre conditii nu este indeplinita}$

$\displaystyle\texttt{deci vom face schimbarea de variabila.}\\x=-t\Rightarrow dx=-dt\\I=\int_{-2}^2\dfrac{x^2}{e^x+1}dx=-\int_{2}^{-2}\dfrac{(-t)^2}{e^{-t}+1}dt=\int_{-2}^2\dfrac{t^2}{\dfrac{1}{e^t}+1}dt=\int_{-2}^2\dfrac{t^2\cdot e^t}{e^t+1}dt\\\texttt{Mai departe adunam integrala initiala cu cea finala (putem }\\\texttt{face asta, deoarece capetele sunt egale ).}\\I+I=\int_{-2}^2\dfrac{x^2}{e^x+1}dx+\int_{-2}^2\dfrac{x^2\cdot e^x}{e^x+1}dx=\int_{-2}^2\dfrac{x^2+x^2\cdot e^x}{e^x+1}dx=$

$\displaystyle=\int_{-2}^2\dfrac{x^2(e^x+1)}{e^x+1}dx=\int_{-2}^2x^2dx=2\cdot\int_0^2x^2dx=\dfrac{2}{3}\cdot x^3|_0^2=\dfrac{16}{3}\\\texttt{Am obtinut deci:}\\2I=\dfrac{16}{3}\Rightarrow \boxed{I=\dfrac{8}{3}}$