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Question

Prove that : 10tan1xdx=π412log2.

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Solution

Suppose that
I=10tan1xdx....(1)
By putting x=tanθ
dx=sec2θ.dθ
When x=0 then θ=0
When x=1 then θ=π4
By equation (1)
I=π40tan1(tanθ)sec2θ.dθ
=π40θ.sec2θ.dθ
=[θ.sec2θdθ]π40π40[ddθ.θsec2θdθ]dθ
=[θ.tanθ]π40π40tanθdθ
=[θ.tanθ]π40+[logcosθ]π40
=π4,tanπ40+logcosπ4logcos0
=π4×1+log12log(1)
=π4+log(1)log2log(1)
=π4log(2)12
=π412log2.

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