[tex]f(x)=x+\frac{2}{x+1}[/tex]
a)
Folosim tabelul integralelor (vezi atasament)
[tex]\int\limits^1_0 {x+\frac{2}{x+1}-x } \, dx =\int\limits^1_0 {\frac{2}{x+1} } \, dx=2ln(x+1)|_0^1=2ln2-2ln1=2ln2[/tex]
b)
[tex]\int\limits^e_1 {(x+\frac{2}{x+2}-\frac{2}{x+1} }) lnx\, dx =\int\limits^e_1 {xlnx} \, dx[/tex]
Calculam prin integrarea prin parti
[tex]f=lnx\ \ \ \ \ \ \ \ f'=\frac{1}{x} \\\\g'=x\ \ \ \ \ \ \ \ \ g=\frac{x^2}{2}[/tex]
[tex]\int\limits^e_1 {xlnx} \, dx=\frac{x^2}{2}lnx|_1^e-\int\limits^e_1 {\frac{x}{2} } \, dx = \frac{x^2}{2}lnx|_1^e-\frac{x^2}{4} |_1^e=\frac{e^2}{2}-\frac{e^2}{4}+\frac{1}{4} =\frac{e^2+1}{4}[/tex]
Un exercitiu similar cu integrale gasesti aici: https://brainly.ro/tema/1026361
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