Bài 3.52 trang 182 SBT giải tích 12

Đề bài

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

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}$

Chọn C.

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