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8. Application and integration — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths8. Application and integration1 Mark
MCQ
A function $y = f (x)$ satisfies the differential equation $\frac{{dy}}{{dx}} - y = \cos x - \sin x$ with initial condition that $y$ is bounded when $x \rightarrow \infty .$ The area enclosed by $y = f (x), y = \cos x$ and the $y-$ axis is
  • ✓
    $\sqrt 2 \; - \;1$
  • B
    $\sqrt 2$
  • C
    $1$
  • D
    $\frac{1}{{\sqrt 2 }}$
✓

Answer

Correct option: A.
$\sqrt 2 \; - \;1$
a
$I.F. = e^{-x}$
$ye^{-x} = \int {{e^{ - x}}(\cos x - \sin x)\,dx} $  put $- x = t$ 
$= - \int {{e^t}(\cos t + \sin t)\,dt} $ 
$= - e^t sin t + c$ 
$y e^{-x} = e^{-x} sin x + c$ 
since $y$ is bounded when $x \rightarrow \infty \Rightarrow c = 0$ 
$y = \sin x$ 
Area $= \int\limits_0^{\frac{\pi }{4}} {(\cos x - \sin x)\,dx} = \sqrt 2 \; - \;1$

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