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Question

Let f(x) be a continuous and not a constant function of all x in its domain, such that
(f(x))2=x0f(t)4sin2t4sin2t+4dt and f(0)=0, then

A
f(3π4)=log(12)
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B
f(π4)=log(32)
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C
f(π4)=log(54)
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D
f(π2)=2
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Solution

The correct option is B f(π4)=log(32)
(f(x))2=x0f(t)4sin 2t4 sin2 t+4dt,f(0)=0

sin 2t=2 tan t1+tan2 t4 sin2 t+4=4(1sin2 t)=4 cos2 t=4sec2 t
(f(x))2=x0f(t)42 tan tsec2 t+4sec2 tdt
(f(x))2=x0f(t)2 sec2 t2+ tan tdt
Taking derivative
2f(x)×f(x)=f(x)×2 sec2 x2+tan x
f(x)=sec2 x2+tan xdx
On integrating both the sides, we have -
f(x)dx=sec2 x2+tan xdx
Let tan x=t sec2 x dx=dt
f(x)=log|2+tan x|+c
f(0)=0c=log 2.
f(x)=log|2+tan x|log 2
f(π4)=log 3log 2=log32
f(π2)2
f(3π4)=log|21|log 2=log12

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