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Question Number 212024 by Spillover last updated on 27/Sep/24

Answered by Frix last updated on 27/Sep/24

$$\underset{0}{\stackrel{\frac{\pi }{4}}{\int }}{\left(\tan x\right)}^{\frac{2}{3}}dx\stackrel{[t={\left(\tan x\right)}^{-\frac{1}{3}}]}{=}3\underset{1}{\stackrel{\mathrm{\infty }}{\int }}\frac{dt}{t^6+1}$$$$\mathrm{Now}\mathrm{decompose}\mathrm{etc}.$$$$\mathrm{I}\mathrm{get}\frac{\pi }{2}+(\ln (\sqrt{3}-1)-\frac{\ln 2}{2})\sqrt{3}$$$$[\underset{0}{\stackrel{\frac{\pi }{2}}{\int }}{\left(\tan x\right)}^{\frac{2}{3}}dx=\pi ]$$

Commented by Frix last updated on 27/Sep/24

$$\alpha >1:\underset{0}{\stackrel{\frac{\pi }{2}}{\int }}{\left(\tan x\right)}^{\alpha }dx=\frac{\pi }{2\sin \frac{\pi }{2\alpha }}$$

Answered by Spillover last updated on 27/Sep/24

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