Bài 3.52 trang 182 SBT giải tích 12

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. \(\displaystyle  \int\limits_0^\pi  {\left| {\sin \left( {x + \frac{\pi }{4}} \right)} \right|dx}  = 2\int\limits_0^{\frac{\pi }{4}} {\sin \left( {x + \frac{\pi }{4}} \right)dx} \)

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 \(\displaystyle  \int\limits_0^\pi  {\left| {\sin \left( {x + \frac{\pi }{4}} \right)} \right|dx} \)\(\displaystyle   = \int\limits_0^{\frac{{3\pi }}{4}} {\sin \left( {x + \frac{\pi }{4}} \right)dx}  - \int\limits_{\frac{{3\pi }}{4}}^\pi  {\sin \left( {x + \frac{\pi }{4}} \right)dx} \)

Chọn C.