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Câu hỏi: Tìm khẳng định đúng trong các khẳng định sau:
A. \(\displaystyle \int\limits_0^\pi {\left| {\sin \left( {x + \frac{\pi }{4}} \right)} \right|dx} = \int\limits_0^\pi {\left| {\sin \left({x - \frac{\pi }{4}} \right)} \right|dx} \)
B. \(\displaystyle \int\limits_0^\pi {\left| {\sin \left( {x + \frac{\pi }{4}} \right)} \right|dx} = \int\limits_0^\pi {\left| {\cos \left({x + \frac{\pi }{4}} \right)} \right|dx} \)
C. \(\displaystyle \int\limits_0^\pi {\left| {\sin \left( {x + \frac{\pi }{4}} \right)} \right|dx} \) \(\displaystyle = \int\limits_0^{\frac{{3\pi }}{4}} {\sin \left( {x + \frac{\pi }{4}} \right)dx} - \int\limits_{\frac{{3\pi }}{4}}^\pi {\sin \left({x + \frac{\pi }{4}} \right)dx} \)
D. \(\displaystyle \int\limits_0^\pi {\left| {\sin \left( {x + \frac{\pi }{4}} \right)} \right|dx} = 2\int\limits_0^{\frac{\pi }{4}} {\sin \left({x + \frac{\pi }{4}} \right)dx} \)
A. \(\displaystyle \int\limits_0^\pi {\left| {\sin \left( {x + \frac{\pi }{4}} \right)} \right|dx} = \int\limits_0^\pi {\left| {\sin \left({x - \frac{\pi }{4}} \right)} \right|dx} \)
B. \(\displaystyle \int\limits_0^\pi {\left| {\sin \left( {x + \frac{\pi }{4}} \right)} \right|dx} = \int\limits_0^\pi {\left| {\cos \left({x + \frac{\pi }{4}} \right)} \right|dx} \)
C. \(\displaystyle \int\limits_0^\pi {\left| {\sin \left( {x + \frac{\pi }{4}} \right)} \right|dx} \) \(\displaystyle = \int\limits_0^{\frac{{3\pi }}{4}} {\sin \left( {x + \frac{\pi }{4}} \right)dx} - \int\limits_{\frac{{3\pi }}{4}}^\pi {\sin \left({x + \frac{\pi }{4}} \right)dx} \)
D. \(\displaystyle \int\limits_0^\pi {\left| {\sin \left( {x + \frac{\pi }{4}} \right)} \right|dx} = 2\int\limits_0^{\frac{\pi }{4}} {\sin \left({x + \frac{\pi }{4}} \right)dx} \)
Phương pháp giải
Xét trong đoạn \(\displaystyle \left[ {0;\pi } \right]\), xét dấu của \(\displaystyle \sin \left( {x + \frac{\pi }{4}} \right)\) và phá dấu giá trị tuyệt đối.
Lời giải chi tiết
Ta có: \(\displaystyle \sin \left( {x + \frac{\pi }{4}} \right) \ge 0\) \(\displaystyle \Leftrightarrow 0 \le x + \frac{\pi }{4} \le \pi \) \(\displaystyle \Leftrightarrow - \frac{\pi }{4} \le x \le \frac{{3\pi }}{4}\).
\(\displaystyle \sin \left( {x + \frac{\pi }{4}} \right) < 0\) \(\displaystyle \Leftrightarrow \pi < x + \frac{\pi }{4} < 2\pi \)\(\displaystyle \Leftrightarrow \frac{{3\pi }}{4} < x < \frac{{7\pi }}{4}\)
Khi đó \(\displaystyle \int\limits_0^\pi {\left| {\sin \left( {x + \frac{\pi }{4}} \right)} \right|dx} \)\(\displaystyle = \int\limits_0^{\frac{{3\pi }}{4}} {\sin \left( {x + \frac{\pi }{4}} \right)dx} - \int\limits_{\frac{{3\pi }}{4}}^\pi {\sin \left({x + \frac{\pi }{4}} \right)dx} \)
Xét trong đoạn \(\displaystyle \left[ {0;\pi } \right]\), xét dấu của \(\displaystyle \sin \left( {x + \frac{\pi }{4}} \right)\) và phá dấu giá trị tuyệt đối.
Lời giải chi tiết
Ta có: \(\displaystyle \sin \left( {x + \frac{\pi }{4}} \right) \ge 0\) \(\displaystyle \Leftrightarrow 0 \le x + \frac{\pi }{4} \le \pi \) \(\displaystyle \Leftrightarrow - \frac{\pi }{4} \le x \le \frac{{3\pi }}{4}\).
\(\displaystyle \sin \left( {x + \frac{\pi }{4}} \right) < 0\) \(\displaystyle \Leftrightarrow \pi < x + \frac{\pi }{4} < 2\pi \)\(\displaystyle \Leftrightarrow \frac{{3\pi }}{4} < x < \frac{{7\pi }}{4}\)
Khi đó \(\displaystyle \int\limits_0^\pi {\left| {\sin \left( {x + \frac{\pi }{4}} \right)} \right|dx} \)\(\displaystyle = \int\limits_0^{\frac{{3\pi }}{4}} {\sin \left( {x + \frac{\pi }{4}} \right)dx} - \int\limits_{\frac{{3\pi }}{4}}^\pi {\sin \left({x + \frac{\pi }{4}} \right)dx} \)
Đáp án C.