Tìm khẳng định sai trong các khẳng định sau:
A. \(\int \limits_0^1 \sin \left( {1 - x} \right)dx = \mathop \smallint \limits_0^1 \sin xdx\)
B. \(\int \limits_0^\pi \sin \frac{x}{2}dx = 2\mathop \smallint \limits_0^{\frac{\pi }{2}} \sin xdx\)
C. \(\int \limits_0^1 {\left( {1 + x} \right)^x}dx = 0\)
D. \(\int \limits_{ - 1}^1 {x^{2007}}\left( {1 + x} \right)dx = \frac{2}{{2009}}\)
Đáp án A:
Đặt \(t = 1 - x \Rightarrow dt = - dx\)
\( \Rightarrow \int \limits_0^1 \sin \left( {1 - x} \right)dx = \int \limits_1^0 \sin t\left( { - dt} \right)\)
\(\int \limits_0^1 \sin \left( {1 - x} \right)dx = \int \limits_0^1 \sin tdt = \int \limits_0^1 \sin xdx\) n
Nên A đúng.
Đáp án B:
Ta có:
\(\int\limits_2^\pi {\sin \frac{x}{2}dx = - 2\left. {\cos \frac{x}{2}} \right|_0^\pi = 2} \)
\(2\int\limits_2^{\frac{\pi }{2}} {\sin xdx = - 2\left. {\cos x} \right|_0^{\frac{\pi }{2}} = 2} \)
Nên \(\int \limits_0^\pi \sin \frac{x}{2}dx = 2\int \limits_0^{\frac{\pi }{2}} \sin xdx\) h
Hay B đúng.
Đáp án D:
\(\begin{array}{l}
\int \limits_{ - 1}^1 {x^{2007}}\left( {1 + x} \right)dx\\
= \int \limits_{ - 1}^1 \left( {{x^{2007}} + {x^{2008}}} \right)dx\\
= \left. {\left( {\frac{{{x^{2008}}}}{{2008}} + \frac{{{x^{2009}}}}{{2009}}} \right)} \right|_{ - 1}^1\\
= \frac{1}{{2008}} + \frac{1}{{2009}} - \frac{1}{{2008}} + \frac{1}{{2009}}\\
= \frac{2}{{2009}}
\end{array}\)
Hay D đúng.
Chọn C.
-- Mod Toán 12
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