Bài tập 3.52 trang 182 SBT Toán 12
Bài tập 3.52 trang 182 SBT Toán 12
Tìm khẳng định đúng trong các khẳng định sau:
A. \(\int \limits_0^\pi \left| {\sin \left( {x + \frac{\pi }{4}} \right)} \right|dx = \int \limits_0^\pi \left| {\sin \left( {x - \frac{\pi }{4}} \right)} \right|dx\)
B. \(\int \limits_0^\pi \left| {\sin \left( {x + \frac{\pi }{4}} \right)} \right|dx = \int \limits_0^\pi \left| {\cos \left( {x + \frac{\pi }{4}} \right)} \right|dx\)
C. \(\begin{array}{l}
\int \limits_0^\pi \left| {\sin \left( {x + \frac{\pi }{4}} \right)} \right|dx\\
= \int \limits_0^{\frac{{3\pi }}{4}} \sin \left( {x + \frac{\pi }{4}} \right)dx - \int \limits_{\frac{{3\pi }}{4}}^\pi \sin \left( {x + \frac{\pi }{4}} \right)dx
\end{array}\)
D. \(\int \limits_0^\pi \left| {\sin \left( {x + \frac{\pi }{4}} \right)} \right|dx = 2\int \limits_0^{\frac{\pi }{4}} \sin \left( {x + \frac{\pi }{4}} \right)dx\)
Ta có:
\(\begin{array}{l}
\sin \left( {x + \frac{\pi }{4}} \right) \ge 0\\
\Leftrightarrow 0 \le x + \frac{\pi }{4} \le \pi \\
\Leftrightarrow - \frac{\pi }{4} \le x \le \frac{{3\pi }}{4}
\end{array}\)
\(\begin{array}{l}
\sin \left( {x + \frac{\pi }{4}} \right) < 0 \Leftrightarrow \pi < x + \frac{\pi }{4} < 2\pi \\
\Leftrightarrow \frac{{3\pi }}{4} < x < \frac{{7\pi }}{4}
\end{array}\)
Khi đó \(\int \limits_0^\pi \left| {\sin \left( {x + \frac{\pi }{4}} \right)} \right|dx\)
\( = \int \limits_0^{\frac{{3\pi }}{4}} \sin \left( {x + \frac{\pi }{4}} \right)dx - \int \limits_{\frac{{3\pi }}{4}}^\pi \sin \left( {x + \frac{\pi }{4}} \right)dx\)
Chọn C.
-- Mod Toán 12
Copyright © 2021 HOCTAP247