Question

Integrala din xpatrat/(xpatrat + 2x + 1)

Answer 1

$\displaystyle=\int \dfrac{x^2+2x+1-2x-1}{x^2+2x+1}dx= \\ \\ \\ =\int \dfrac{x^2+2x+1}{x^2+2x+1}dx-\int \dfrac{2x+2-1}{x^2+2x+1}= \\ \\ \\ =\int dx-\int \dfrac{2(x+1)}{(x+1)^2}dx+\int \dfrac{1}{(x+1)^2}dx \\ \\ \\ =\int dx-2\int \dfrac{1}{x+1}dx+\int \dfrac{1}{(x+1)^2}dx \\ \\ \\ =x-2\ln(x+1)-\dfrac{1}{x+1}.$