Khẳng định nào sau đây sai?
A. \(\int \limits_{\frac{\pi }{2}}^\pi \frac{{\sin x}}{x}dx < \int \limits_{\frac{\pi }{2}}^\pi \frac{{\cos x}}{x}dx\)
B. \(\int \limits_{\frac{\pi }{4}}^1 \frac{{\tan x}}{x}dx > \int \limits_{\frac{\pi }{4}}^1 \frac{{\cot x}}{x}dx\)
C. \(\int \limits_0^{\frac{\pi }{4}} {\sin ^4}xdx < \int \limits_0^{\frac{\pi }{2}} dx\)
D. \(\int \limits_1^e \frac{{\ln x}}{x}dx < \int \limits_1^e \frac{{{e^x}}}{x}dx\)
Xét \(I = \int \limits_{\frac{\pi }{2}}^\pi \frac{{\sin x}}{x}dx - \int \limits_{\frac{\pi }{2}}^\pi \frac{{\cos x}}{x}dx\)
\( = \int \limits_{\frac{\pi }{2}}^\pi \left( {\frac{{\sin x - \cos x}}{x}} \right)dx\)
Dễ thấy trên đoạn \(\left[ {\frac{\pi }{2};\pi } \right]\)
thì x > 0 và \(\sin x > 0 > \cos x\)
\( \Rightarrow \sin x - \cos x > 0\)
Suy ra \(\frac{{\sin x - \cos x}}{x} > 0\)
\( \Rightarrow I = \int \limits_{\frac{\pi }{2}}^\pi \left( {\frac{{\sin x - \cos x}}{x}} \right)dx > 0\)
\( \Rightarrow \int \limits_{\frac{\pi }{2}}^\pi \frac{{\sin x}}{x}dx > \int \limits_{\frac{\pi }{2}}^\pi \frac{{\cos x}}{x}dx\)
Vậy A sai.
Chọn A.
-- Mod Toán 12
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