Question

Puteti sa ma ajutati si pe mine cu rezolvarea la acest exercitiu: $\int\limits {\frac{3x-5}{x^{2}-3x+2 } } \, dx =$

Answer 1

Răspuns:

$\frac{3x-5}{x^{2}-3x+2}=\frac{3x-5}{(x-1)(x-2)}=\frac{A}{x-1}+\frac{B}{x-2}=\frac{Ax-2A+Bx-B}{(x-1)(x-2)}=\frac{(A+B)x-(2A+B)}{(x-1)(x-2)} ,~deci~\left \{ {{A+B=3} \atop {2A+B=5}} \right. ~deci~A=2;~B=1.~Atunci  \frac{3x-5}{x^{2}-3x+2}=\frac{2}{x-1} +\frac{1}{x-2} .Revenim~la~integrala\\\int {\frac{3x-5}{x^{2}-3x+2}} \, dx =\int {\frac{2}{x-1} } \, dx +\int {\frac{1}{x-2} } \, dx=2*ln|x-1|+ln|x-2|+C=//=ln|(x-1)^{2}(x-2)|+C.$

Explicație pas cu pas: