Question Number 148376 by ArielVyny last updated on 27/Jul/21

∫_1 ^∞ x^i lnxdx i^2 =−1

Answered by mathmax by abdo last updated on 27/Jul/21

I=∫_1 ^∞ x^i logx dx cha7gement logx=t give x=e^t ⇒ I=∫_0 ^∞ (e^t )^i t e^(t ) dt=∫_0 ^∞ t e^(it+t) dt =∫_0 ^∞ t e^((1+i)t) dt =_((1+i)t=z) ∫_0 ^∞ (z/(1+i))e^z (dz/(1+i))=(1/((1+i)^2 ))∫_0 ^∞ ze^z dz by parts but ∫_0 ^∞ ze^(z ) dz is divergent...!

Commented byArielVyny last updated on 28/Jul/21

thank sir