1/ 2/ (cos(1/x))/(x2) dx = …

Official answer: -1. Explanation: We need to evaluate the definite integral: I = 1/ 2/ (cos(1/x))/(x2) dx Substitute t = (1)/(x): dt = -(1)/(x2) dx (dx)/(x2) = -dt Transforming the limits of integration: - When x = (1)/( ), t = - When x = (2)/( ), t = ( )/(2) Substituting into the integral: I = /2 cos(t) (-dt) = /2 cos(t) dt I = [sin(t)] /2 = sin( ) - sin (( )/(2) ) = 0 - 1 = -1

Calculus

GATE CSE 2015 Set 1 · Q44

NAT2 marksEasy

Time00:00

$\int_{1/\pi}^{2/\pi} \frac{\cos(1/x)}{x^2} dx =$ _________.

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Official answer: -1

Calculus

We need to evaluate the definite integral:

I = \int_{1/\pi}^{2/\pi} \frac{\cos(1/x)}{x^2} dx

Substitute $t = \frac{1}{x}$ :

dt = -\frac{1}{x^2} dx \implies \frac{dx}{x^2} = -dt

Transforming the limits of integration:

When $x = \frac{1}{\pi}$ , $t = \pi$
When $x = \frac{2}{\pi}$ , $t = \frac{\pi}{2}$

Substituting into the integral:

I = \int_{\pi}^{\pi/2} \cos(t) (-dt) = \int_{\pi/2}^{\pi} \cos(t) dt

I = [\sin(t)]_{\pi/2}^{\pi} = \sin(\pi) - \sin\left(\frac{\pi}{2}\right) = 0 - 1 = -1

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This Calculus previous year question is Q44 of the GATE CSE 2015 Set 1 paper. Find more GATE CSE PYQs on Calculus in Engineering Mathematics, or browse all GATE CSE PYQs.

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Calculus in GATE CSE

Calculus has 40 GATE CSE PYQs across 25 papers from 2011 to 2026. 4 of them are 2-mark NAT questions like this one, and GATE CSE 2015 Set 1 has 4.

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Independent practice explanation verified by the GateAI team. GATE is conducted by the IITs; question text follows the official paper.