Question-151145

Question Number 151145 by mnjuly1970 last updated on 18/Aug/21

Answered by qaz last updated on 18/Aug/21

\int_{0}^{\infty} \frac{x^{2}}{\cosh^{2} (x^{2})} dx

= 2 \int_{0}^{\infty} \frac{x^{2}}{1 + \cosh ({2 x}^{2})} dx

= \int_{0}^{\infty} \frac{\sqrt{x}}{1 + \cosh (2 x)} dx

= \int_{0}^{\infty} \frac{2 \sqrt{x} e^{2 x}}{{(1 + e^{2 x})}^{2}} dx

= - \frac{\sqrt{x}}{1 + e^{2 x}} ∣_{0}^{\infty} + \frac{1}{2} \int_{0}^{\infty} \frac{x^{- 1 / 2}}{1 + e^{2 x}} dx

= \frac{1}{2} \int_{0}^{\infty} \frac{x^{- 1 / 2} e^{- 2 x}}{1 + e^{- 2 x}} dx

= \frac{1}{2} \underset{n = 0}{\sum^{\infty}} {(- 1)}^{n} \int_{0}^{\infty} x^{- 1 / 2} e^{- (2 n + 2) x} dx

= \frac{\sqrt{π}}{2} \underset{n = 0}{\sum^{\infty}} \frac{{(- 1)}^{n}}{{(2 n + 2)}^{\frac{1}{2}}}

= \frac{\sqrt{π}}{2 \sqrt{2}} \underset{n = 1}{\sum^{\infty}} \frac{{(- 1)}^{n - 1}}{n^{\frac{1}{2}}}

= \frac{\sqrt{π}}{2 \sqrt{2}} η (\frac{1}{2})

= \frac{\sqrt{π}}{2 \sqrt{2}} (1 - 2^{1 - \frac{1}{2}}) ζ (\frac{1}{2})

\Rightarrow k = \frac{\sqrt{π}}{2 \sqrt{2}} (1 - \sqrt{2})

Commented by mnjuly1970 last updated on 18/Aug/21

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