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Question Number 151780 by Tawa11 last updated on 23/Aug/21

Answered by puissant last updated on 23/Aug/21

Q=∫_0 ^1 ((arctanx)/x)dx arctanx=Σ_(n=0) ^∞ (−1)^n (x^(2n+1) /(2n+1)) ⇒ ((arctanx)/x)=Σ_(n=0) ^∞ (((−1)^n x^(2n) )/(2n+1)) ⇒ ∫_0 ^1 ((arctanx)/x)dx=Σ_(n=0) ^∞ (((−1)^n )/(2n+1))∫_0 ^1 x^(2n) dx = Σ_(n=0) ^∞ (((−1)^n )/((2n+1)^2 ))

Q=∫01arctanxxdxarctanx=∑∞n=0(−1)nx2n+12n+1⇒arctanxx=∑∞n=0(−1)nx2n2n+1⇒∫01arctanxxdx=∑∞n=0(−1)n2n+1∫01x2ndx=∑∞n=0(−1)n(2n+1)2

Commented by Tawa11 last updated on 23/Aug/21

Thanks sir. God bless you.

Thankssir.Godblessyou.

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