[tex]f(x)=\sqrt{x^{2}+2}[/tex]
a)
Vezi tabelul de integrale din atasament
[tex]\int\limits^3_0 {x^2+2} \, dx =\frac{x^3}{3}|_0^3+2x|_0^3=9+6=15[/tex]
b)
[tex]I_n=\int\limits^1_0 {x^n\sqrt{x^2+2} } \, dx \\\\0\leq I_n\leq \int\limits^1_0{x^n\sqrt{3} } \, dx \\\\x\in[0,1]\\\\\int\limits^1_0{x^n\sqrt{3} } \, dx =\sqrt{3} \frac{x^{n+1}}{n+1} |_0^1=\frac{\sqrt{3} }{n+1}[/tex]
[tex]\lim_{n \to +\infty} I_n= \lim_{n \to +\infty} \frac{\sqrt{3} }{n+1} =0[/tex]
c)
[tex]I_n=\int\limits^1_0 {x^n\sqrt{x^2+2} } \, dx \\\\I_n=\int\limits^1_0 {x^n\frac{x^2+2}{\sqrt{x^2+2} } } \, dx =\int\limits^1_0{x^{n+1}\frac{x}{\sqrt{x^2+2} } } \, dx +2\int\limits^1_0{x^{n-1}\frac{x}{\sqrt{x^2+2} } } \, dx \\\\I_n=\sqrt{3}-(n+1)I_n+2\sqrt{3}-2(n-1)I_{n-2}\\\\ (n+2)I_n+2(n-1)I_{n-2}=3\sqrt{3}[/tex]
Un alt exercitiu cu integrale gasesti aici: https://brainly.ro/tema/9919077
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