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Question Number 42506 by maxmathsup by imad last updated on 26/Aug/18
find the value of ∫_0 ^∞ ((ln(t))/((^3 (√t^2 ))(1+t)))dt .
findthevalueof∫0∞ln(t)(3t2)(1+t)dt.
Commented by maxmathsup by imad last updated on 29/Aug/18
we have proved hat ∫_0 ^∞ ((ln(t)t^(a−1) )/(1+t))dt =−π^2 ((cos(πa))/(sin^2 (πa))) and ∫_0 ^∞ ((ln(t))/((^3 (√t^2 ))(1+t)))dt = ∫_0 ^∞ ((ln(t).t^(−(2/3)) )/(1+t))dt =∫_0 ^∞ ((ln(t) t^((1/3)−1) )/(1+t))dt =−π^2 ((cos((π/3)))/(sin^2 ((π/3)))) =−π^2 (1/(2(((√3)/2))^2 )) =((−π^2 )/((√3)/2)) =((−2π^2 )/( (√3))) .
wehaveprovedhat∫0∞ln(t)ta−11+tdt=−π2cos(πa)sin2(πa)and∫0∞ln(t)(3t2)(1+t)dt=∫0∞ln(t).t−231+tdt=∫0∞ln(t)t13−11+tdt=−π2cos(π3)sin2(π3)=−π212(32)2=−π232=−2π23.

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