门| GATE-CS-2015（套装1）|第 54 题

$\int_{\frac{1}{\pi}}^{\frac{2}{\pi}} cos(\frac{1/x}{x^{2}})dx = ...........$
(一) 0
(乙) -1
(三) 1
(四)无限答案：(乙)
解释：
令 f(x) 为给定的函数。我们假设 \[\frac{1}{x} = z\]

两边微分，我们得到

    $\[\frac{-1}{x^2} dx = dz\] Now, accordingly, the lower limit of the integral is \[ z = \frac{1}{\frac{1}{\pi}} = \pi\] and the upper limit for the integral is \[ z = \frac{1}{\frac{2}{\pi}} = \frac{\pi}{2}\] So, the given function now becomes \[ f(x)= - \int_\pi^{\frac{\pi}{2}} cos(z) dz \] \[ f(x)= \int_\frac{\pi}{2}^{\pi} cos(z) dz \] \[f(x) = sin(z) ,\] and the upper limit is π and the lower limit is π/2 So, \[f(x) = sin(\pi) - sin(\frac{\pi}{2})\] \[f(x) = 0 - 1\] \[f(x) = -1\] So, the required answer is -1.$

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