[tex]f(x)=x^{2}+\ln x[/tex]
a)
Vezi tabelul de integrale din atasament
[tex]\int\limits^3_1 {x^2} \, dx =\frac{x^3}{3}|_1^3=\frac{27}{3}-\frac{1}{3} =\frac{26}{3}[/tex]
b)
[tex]\int\limits^2_1 {lnx} \, dx =[/tex]
Integram prin parti
[tex]f=lnx\ \ \ \ f'=\frac{1}{x} \\\\g'=1\ \ \ \ g=x[/tex]
[tex]\int\limits^2_1 {lnx} \, dx =xlnx|_1^2-\int\limits^2_1 {1 \, dx =2ln2-ln1-x|_1^2=2ln2-2+1=2ln2-1[/tex]
c)
[tex]f(\frac{1}{x})=\frac{1}{x^2}+ln\frac{1}{x} = \frac{1}{x^2}+ln1-lnx=\frac{1}{x^2}-lnx[/tex]
[tex]\int\limits^2_1 {\frac{1}{x^3}-\frac{lnx}{x} } \, dx =\int\limits^2_1 {\frac{1}{x^3}} \, dx -\int\limits^2_1 {\frac{lnx}{x} } \, dx =\frac{x^{-2}}{-2} |_1^2-\int\limits^2_1 {lnx\cdot ln'x } \, dx=-\frac{1}{8}+\frac{1}{2}-\frac{1}{2}ln^2x|_1^2=-\frac{1}{8}+\frac{1}{2}-\frac{1}{2}ln^22=\frac{3-4ln^22}{8}[/tex]
Un alt exercitiu cu integrale gasesti aici: https://brainly.ro/tema/9919079
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