show-that-0-1-lnx-x-2-1-dx-2-8-

Question Number 210098 by klipto last updated on 30/Jul/24

show that

\int_{0}^{1} \frac{lnx}{x^{2} - 1} dx = \frac{π^{2}}{8}

Commented by AlagaIbile last updated on 30/Jul/24

\frac{1}{x^{2} - 1} = - \underset{n = 0}{\sum^{\infty}} x^{2 n}

\Rightarrow - \underset{n = 0}{\sum^{\infty}} \int_{0}^{1} x^{2 n} \ln x d x = \underset{n = 0}{\sum^{\infty}} \frac{1}{{(2 n + 1)}^{2}}

= \frac{1}{1^{2}} + \frac{1}{3^{2}} + \frac{1}{5^{2}} + \frac{1}{7^{2}} + \dots

= [\frac{1}{1^{2}} + \frac{1}{2^{2}} + \frac{1}{3^{2}} + \frac{1}{4^{2}} + \dots] - \frac{1}{2^{2}} [\frac{1}{1^{2}} + \frac{1}{2^{2}} + \frac{1}{3^{2}} + \frac{1}{4^{2}} + \dots]

= \frac{3}{4} [\frac{1}{1^{2}} + \frac{1}{2^{2}} + \frac{1}{3^{2}} + \frac{1}{4^{2}} + \cdot \cdot \cdot]

= \frac{3}{4} \frac{π^{2}}{6}

= \frac{π^{2}}{8}

Commented by klipto last updated on 02/Aug/24

great bigman can we chat on whatapp

Commented by AlagaIbile last updated on 02/Aug/24

N o p r o b l e m . + 2348131291501

Commented by klipto last updated on 02/Aug/24

alrt now my man i go message you

Answered by Spillover last updated on 01/Aug/24

Answered by Spillover last updated on 02/Aug/24

Answered by Spillover last updated on 02/Aug/24

Answered by Spillover last updated on 02/Aug/24

Commented by klipto last updated on 02/Aug/24

can we chat on whatsapp

Commented by Spillover last updated on 02/Aug/24

n o

Answered by Spillover last updated on 02/Aug/24

Commented by klipto last updated on 02/Aug/24

please can we chat on whatsapp ?

Commented by Spillover last updated on 02/Aug/24

n o

Commented by klipto last updated on 02/Aug/24

what about facebook

Commented by Spillover last updated on 02/Aug/24

n o

Commented by klipto last updated on 02/Aug/24

okay

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