₀¹00} e}^{x-[x]} dx is equal to

Correct Answer is : 100(e−1)100(e-1)100(e−1)

Correct Answer is : 100(e−1)100(e-1)100(e−1)

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CGPET 2015 Solved Paper

Section: Mathematics

Question : 139 of 150

Marks: +1, -0

\int\limits_{0^{100} e}^{x-[x]} dx

Solution:

Given differential equation is

{dy}/{dx} = y \tan x - y^2 \sec x

⇒

{dy}/{dx} - y \tan x = -y^2 \sec x

⇒

1/y^2 {dy}/{dx} - \tan x/y = -\sec x

Put

-1/y = u

⇒

1/y^2 {dy}/{dx} = {du}/{dx}

...(i)

From Eq. (i),

{du}/{dx} + \tan x.u = - \sec x

...(ii)

This is linear differential equation of the form

{du}/{dx} +P.u = Q, where P= \tan x, Q = -\sec x

∴

IF = e^{\int \tan x \, dx} = e^{\log \sec x} = \sec x

Hence, general solution is

u.IF = \int IF.Q dx +C_1

⇒

u.\sec x = \int(\sec x).(-\sec x) dx + C_1

⇒

u.\sec x = - \int\sec^2 xdx +C_1

⇒

u \sec x = - \tan x + c_1

⇒

{- \sec x}/y = -\tan x +C_1 [∵u= {-1}/y]

⇒

\sec x = y(\tan x -C_1)

⇒

\sec x = y(\tan x + C) [where, C = -C_1]

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