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33 votes
33 votes

\large{\displaystyle \red{ \tt\int_(0)^(1) \frac{x \ln^(2)(x) }{1 + {x}^(2) } dx }}

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User Demolishun
by
2.6k points

1 Answer

14 votes
14 votes

Explanation:


{\displaystyle \red{ \tt = (1)/(2) \int_(0)^(1) \frac{x {}^{ (1)/(2) } \ln^(2)(x {}^{ (1)/(2) } ) }{1 + {x}^{} } \frac{dx}{ {x}^{ (1)/(2) } } }}


{\displaystyle \red{ \tt = (1)/(8) \int_(0)^(1) \frac{\ln^(2)(x)}{ {1 + x}^{ \frac{}{} } }dx }}


{\displaystyle \red{ \tt = (1)/(8) \sum_(k = 0)^( \infty ) ( - 1) {}^(k) \int_(0)^(1) {x}^(k) \ln^(2)(x)dx }}


{\displaystyle \red{ \tt = (1)/(8) \sum_(k = 0)^( \infty ) ( - 1) {}^(k) \frac{( - 1 {)}^(2) 2!}{(k + 1 {)}^(3) }}}


{\displaystyle \red{ \tt = (1)/(4) \sum_(k = 1)^( \infty ) \frac{( - 1 {)}^(k - 1)}{k{}^(3) }}}


\tt \red{ = (1)/(4)\eta(3) }=


\tt \red{ = (3)/(16) \zeta(3) }

User Lennart
by
3.0k points
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