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Differential Equations question

2022 · 25 Jun · Shift 1 · Q32

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JEE Main
Mathematics
Differential Equations
2022 · 25 Jun · Shift 1 · Q32

Differential Equations question

2022 · 25 Jun · Shift 1 · Q32

JEE MainMathematicsDifferential EquationsMCQ+4 / −1

Let

g:(0,\infty ) \to R

be a differentiable function such that

\int {\left( {{{x(\cos x - \sin x)} \over {{e^x} + 1}} + {{g(x)\left( {{e^x} + 1 - x{e^x}} \right)} \over {{{({e^x} + 1)}^2}}}} \right)dx = {{x\,g(x)} \over {{e^x} + 1}} + c}

, for all x > 0, where c is an arbitrary constant. Then :

A
g is decreasing in $\left( {0,{\pi \over 4}} \right)$
B
g' is increasing in $\left( {0,{\pi \over 4}} \right)$
C
g + g' is increasing in $\left( {0,{\pi \over 2}} \right)$
D
g $-$ g' is increasing in $\left( {0,{\pi \over 2}} \right)$

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