0-1-1-a-x-a-x-2-dx-1-ln-a-ln-3-pi-3-3-

Question Number 186911 by Spillover last updated on 11/Feb/23

\int_{0}^{\infty} \frac{1}{1 + a^{x} + a^{\frac{x}{2}}} d x = \frac{1}{\ln a} [\ln 3 - \frac{π}{3 \sqrt{3}}]

Answered by witcher3 last updated on 13/Feb/23

a^{\frac{x}{2}} = t, a > 1

x = 2 \frac{\ln (t)}{\ln (a)}

dx = \frac{2}{\ln (a) t} dt

\Leftrightarrow \frac{2}{\ln (a)} \int_{1}^{\infty} \frac{dt}{t (1 + t + t^{2})}

\frac{2}{\ln (a)} \int_{0}^{1} y \frac{dy}{1 + y + y^{2}}

= \frac{2}{\ln (a)} \int_{0}^{1} \frac{y + \frac{1}{2}}{1 + y + y^{2}} - \frac{1}{\ln (a)} \int_{0}^{1} \frac{dy}{{(y + \frac{1}{2})}^{2} + \frac{3}{4}}

= \frac{2}{\ln (a)} . \frac{1}{2} {[\ln (1 + y + y^{2})]}_{0}^{1} - \frac{1}{\ln (a)} \int_{0}^{1} . \frac{4}{3} \frac{dy}{1 + (\frac{2 y}{\sqrt{3}} + \frac{1}{\sqrt{3}})}

\frac{1}{\ln (a)} [\ln (3) - \frac{2}{\sqrt{3}} \tan^{- 1} (\sqrt{3}) + \frac{2}{\sqrt{3}} \tan^{- 1} (\frac{1}{\sqrt{3}}))

= \frac{1}{\ln (a)} [\ln (3) - \frac{π}{3 \sqrt{3}}]

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