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: \(I=2ln2-\frac{5}{4}\).