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Question

Let λ=91(12+14+log3 x)dx+213x2[x]3{x}dx then 2λ equals [where [.] denotes the greatest integer function and {.} denotes the fractional part function]

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Solution

{x}=x[x]λ=91(12+14+log3x )dx+213x2xdxf1(x)=3x2x and f(x)=12+14+log3 xλ=91f(x)dx+21f1(x)dxf1(x)=tx=f(x)dx=f(t)dt(1)λ=91f(x)dx+91tf(t)dtλ=91f(x)dx+(tf(t))9191f(t)dt (using (1))we get, λ=9.f(9)1.f(1)λ=9×21×1=172λ=34

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