b)
[tex]I_{n+1} \leq I_{n} <=> I_{n+1} - I_{n} \leq 0 \\ \\[/tex]
[tex]I_{n+1} - I_{n} = \int\limits^1_0 {xe^{-(n+1)x^{2}} - xe^{-nx^{2}} } \, dx = \\ \\ [/tex]
[tex]\int\limits^1_0 {xe^{-nx^{2}-x^{2}} - xe^{-nx^{2}} } dx \\ \\ = \int\limits^1_0 {xe^{-nx^{2}}*e^{-x^{2}} - xe^{-nx^{2}} } dx \\[/tex]
[tex]= \int\limits^1_0 {xe^{-nx^{2}}(e^{-x^{2}} - 1) }\ dx \\ \\[/tex]
[tex]Pentru \ x \ apartine \ [0,1] \ si \ 'n' \ natural, avem \\[/tex]
[tex]e^{-nx^{2}} > 0 \ si \ e^{-x^2} -1 \leq 0 \\ \\ => I_{n+1} - I_{n} \leq 0[/tex]
c)
[tex]I_{n}= \int\limits^1_0 { \, xe^{-nx^{2}}} dx [/tex]
[tex]= - \frac{1}{2n} \int\limits^1_0 {(e^{-nx^{2}})' dx \\ \\ [/tex]
[tex]=- \frac{1}{2n}e^{-nx^{2}} \ de \ la \ 0 \ la \ 1[/tex]
[tex]= \frac{1}{2n} (1- \frac{1}{e^n} )[/tex]
