Explicație pas cu pas:
Rescriem fractia sub alta forma:
[tex]\frac{2x+1}{2x-1}=\frac{2x-1+2}{2x-1}=\frac{2x-1}{2x-1}+\frac{2}{2x-1}=1+\frac{2}{2x-1}[/tex]
Rescriem integrala:
[tex]\int\ {\frac{2x+1}{2x-1}} \, dx= \int\ {(1+\frac{2}{2x-1})} \, dx=\int\ {1} \, dx+\int\ {\frac{2}{2x-1}} \, dx=x+2*\int\ {\frac{1}{2x-1}} \, dx=x+2*\frac{ln|2x-1|}{2}+C=x+ln|2x-1|+C[/tex]
Ce am aplicat?
- [tex]\int\ {f(x)+g(x)} \, dx=\int\ {f(x)} \, dx+\int\ {g(x)} \, dx[/tex]
- [tex]\int\ {1} \, dx=x+C[/tex]
- [tex]\int\ {\frac{1}{x}} \, dx =ln|x|+C[/tex]
- [tex]\int\ {\frac{1}{ax+b}} \, dx=\frac{1}{a}*ln|ax+b|+C[/tex]
- [tex]\int\ {\alpha f(x)} \, dx=\alpha*\int\ {f(x)} \, dx, \alpha \in IR[/tex]
