Întrebare Integrala definita

$$\int_{1}^{e}\frac{\sqrt{lnx}}{x\left (1+\sqrt{lnx} \right )^3} dx$$

Buna ziua,

Notati integrala de calculat cu $I$.

Amplificati fractia din integrala cu $2\sqrt{lnx}$, asa aveti $I=2\int_{1}^{e}\frac{\sqrt{lnx}^{2}}{2x\sqrt{lnx}\left ( 1+\sqrt{lnx} \right )^{3}}dx$.
Faceti schimbarea de variabila $t=\sqrt{lnx}\Rightarrow dt=\frac{1}{2x\sqrt{lnx}}dx$, capetele: $x=1\Rightarrow t=0$ si $x=e\Rightarrow t=1$.

Aveti de calculat integrala $I=2\int_{0}^{1}\frac{t^{2}}{\left ( 1+t \right )^{3}}$.

Descompuneti functia rationala din integrala in suma de functii rationale simple, obtineti:
$\frac{t^{2}}{\left ( 1+t \right )^{3}}=\frac{1}{1+t}-\frac{2}{\left ( 1+t \right )^{2}}+\frac{1}{\left ( 1+t \right )^{3}}$.

Rezultat: $I=2ln2-\frac{5}{4}$.