IIT JEE , AIEEE Forum - Mathematics , Integral Calculus - Sir please solve this problem please! I haven't been able to solve it myself

rahul_sgh16_in

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₃

^{[log[x]] dx where [.] denotes greatest intereger value function}

edison

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[log[x]] dx = I

^{_{Greatest Integer function can be defined as}}

^{_{f(x) = [x] = -1 for x belongs to [ -1, 0)}}

^{_{0 for x belongs to [ 0,1)}}

^{_{1 for x belongs to [ 1, 2)}}

^{_{Here '[' means closed interval and ')' means open interval thus}}

^{_{x belongs to [ -1, 0) means that -1 belongs to the interval but '0' does not}}

^{_{So, We split above function I as follows}}

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[log[x]] dx = I = I₁ + I₂ + I₃+ I₄+ I₅+ I₆+ I₇

₄

I₁= ₃

[log[x]] dx = [log 3]

₅

I₂= ₄

[log[x]] dx= [log 4]

₆

I₃= ₅

[log[x]] dx = [log 5]

₇

I₄= ₆

[log[x]] dx= [log 6]

₈

I₅= ₇

[log[x]] dx= [log 7]

₉

I₆= ₈

[log[x]] dx= [log 8]

₁₀

I₇= ₉

[log[x]] dx = [log 9]

or I = [log 3] + [log 4] + [log 5] + [log 6] + [log 7] + [log 8] + [log 9]

Here if it is natural log i.e. base is 'e' then

or I = 1 + 1 + 1 + 1 + 1 + 2 + 2 = 9