Question

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

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Answer 1

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) }`