\(H=\int\limits^{\dfrac{\pi}{2}}_0\dfrac{2sinx.cosxdx}{\left(2+sinx\right)^2}\)
Đặt \(2+sinx=t\Rightarrow sinx=t-2\Rightarrow cosx.dx=dt\) ; \(\left\{{}\begin{matrix}x=0\Rightarrow t=2\\x=\dfrac{\pi}{2}\Rightarrow t=3\end{matrix}\right.\)
\(H=\int\limits^3_2\dfrac{2\left(t-2\right)dt}{t^2}=2\int\limits^3_2\left(\dfrac{1}{t}-\dfrac{2}{t^2}\right)dt=2\left(ln\left|t\right|+\dfrac{2}{t}\right)|^3_2=2ln\left(\dfrac{3}{2}\right)-\dfrac{2}{3}\)
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