Tinkutara Equation Editor: Math Forum Question #: 103190


Question Number 103190 by bemath last updated on 13/Jul/20

	Answered by abdomathmax last updated on 13/Jul/20
	$ln(tanx) =t ⇒ I =∫_(π/4) ^(π/2) ln(ln(tanx))dx tanx =e^(t ) ⇒x =arctan(e^t ) ⇒ I =∫_0 ^(+∞) ln(t) ×(e^t /(1+e^(2t) )) dt =∫_0 ^∞ ((e^t ln(t))/(1+e^(2t) ))dt =∫_0 ^∞ ((e^(−t) ln(t))/(1+e^(−2x) )) dt =∫_0 ^∞ e^(−t) ln(t)(Σ_(n=0) ^(∞ ) (−1)^(n ) e^(−(nt) )dt =Σ_(n=0) ^∞ (−1)^(n ) ∫_0 ^∞ e^(−(n+1)t) ln(t)dt A_n =∫_0 ^∞ e^(−(n+1)t) ln(t)dt =_((n+1)t=u) ∫_0 ^∞ e^(−u) ln((u/(n+1)))(du/(n+1)) =(1/(n+1)) ∫_0 ^∞ {e^(−u) ln(u)−e^(−u) ln(n+1))du =−(γ/(n+1))−((ln(n+1))/(n+1)) ∫_0 ^∞ e^(−u ) du =−(γ/(n+1))−((ln(n+1))/(n+1)) ⇒ I =−γ Σ_(n=0) ^(∞ ) (((−1)^n )/(n+1)) −Σ_(n=0) ^∞ (((−1)^n )/(n+1))ln(n+1) Σ_(n=0) ^∞ (((−1)^n )/(n+1)) =Σ_(n=1) ^(∞ ) (((−1)^(n−1) )/n) =ln(2) Σ_(n=0) ^∞ (((−1)^n )/(n+1))ln(n+1) =Σ_(n=1) ^∞ (((−1)^(n−1) )/n)ln(n) ⇒I =−γln(2) +Σ_(n=1) ^∞ (((−1)^n lnn)/n) ...be continued...$
	Commented byabdomathmax last updated on 13/Jul/20