Question

int\ {x/x^2+4} \, dx

integrala din x supra x la puterea a doua plus 4

Va rog rezolvare. Dau coroana rapid.

Answer 1

Metoda schimbarii de variabila.

Answer 2

$\displaystyle \\ \\ \int \dfrac{x}{x^2+4}\, dx = \int \dfrac{1}{x^2+4}\cdot x\, dx \overset{(*)}{=} \\ \\ x^2+4 = t \Rightarrow (2x+0)\, dx = 1\, dt \Rightarrow 2x\, dx = dt \Rightarrow \\ \Rightarrow x \, dx = \dfrac{1}{2}\, dt \\ \\ \overset{(*)}{=} \int \dfrac{1}{t}\cdot \dfrac{1}{2}\, dt = \dfrac{1}{2}\int \dfrac{1}{t}\, dt =\dfrac{1}{2}\cdot \ln(t) + C = \\ \\ = \ln\left(t^{\dfrac{1}{2}}\right) +C= \ln \Big(\sqrt{t}\Big) + C = \ln\Big(\sqrt{x^2+4}\Big)+C$