Toán Lớp 12: Tích phân của $I=\int\limits^{\small\dfrac \pi2}_{\small\dfrac \pi3} \dfrac{dx}{\sin x}$ có giá trị là___?
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Toán Lớp 12: Tích phân của $I=\int\limits^{\small\dfrac \pi2}_{\small\dfrac \pi3} \dfrac{dx}{\sin x}$ có giá trị là___?
Comments ( 1 )
\quad I = \displaystyle\int\limits_{\tfrac{\pi}{3}}^{\tfrac{\pi}{2}}\dfrac{dx}{\sin x}\\
\Leftrightarrow I = \displaystyle\int\limits_{\tfrac{\pi}{3}}^{\tfrac{\pi}{2}}\dfrac{dx}{\dfrac{2\sin\left(\dfrac{x}{2}\right)}{\cos\left(\dfrac{x}{2}\right)}\cdot \cos^2\left(\dfrac{x}{2}\right)}\\
\text{Đặt}\ t=\tan\left(\dfrac{x}{2}\right)\\
\Rightarrow dt = \dfrac{1}{2\cos^2\left(\dfrac{x}{2}\right)}\\
\text{Đổi cận:}\\
\begin{array}{c|ccc}x&\dfrac{\pi}{3}&&\dfrac{\pi}{2}\\\hline
t&\dfrac{\sqrt3}{3}&&1
\end{array}\\
\text{Ta được:}\\
\quad I = \displaystyle\int\limits_{\tfrac{\sqrt3}{3}}^1\dfrac{1}{\tan t}dt\\
\Leftrightarrow I = \ln t\Bigg|_{\tfrac{\sqrt3}{3}}^1\\
\Leftrightarrow I = \ln 1 – \ln\dfrac{\sqrt3}{3}\\
\Leftrightarrow I = \dfrac12\ln3
\end{array}\)