Tex int frac 1 pi frac 2 pi cos( frac 1 x x 2 )dx ........... Tex

GATE | GATE-CS-2015 (Set 1) | Question 54

Last Updated : 14 Feb, 2018

$\int_{\frac{1}{\pi}}^{\frac{2}{\pi}} cos(\frac{1/x}{x^{2}})dx = ...........$
(A) 0
(B) -1
(C) 1
(D) infinite

Answer: (B)

Explanation:
Let f(x) be the given function. We assume that \[\frac{1}{x} = z\]

Differentiating both sides, we get

$\[\frac{-1}{x^2} dx = dz\] 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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