Question Number 128953 by mathmax by abdo last updated on 11/Jan/21
$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{ln}\left(\mathrm{1}+\mathrm{x}^{\mathrm{6}} \right)\mathrm{dx} \\ $$
Answered by Lordose last updated on 12/Jan/21
$$ \\ $$$$\Omega\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \mathrm{ln}\left(\mathrm{1}+\mathrm{x}^{\mathrm{6}} \right)\mathrm{dx}\:=\:\underset{\mathrm{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{\mathrm{n}−\mathrm{1}} }{\mathrm{n}}\int_{\mathrm{0}} ^{\:\mathrm{1}} \mathrm{x}^{\mathrm{6n}} \mathrm{dx} \\ $$$$\Omega\:=\:\underset{\mathrm{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\left(−\mathrm{1}\right)^{\mathrm{n}−\mathrm{1}} }{\mathrm{n}\left(\mathrm{6n}+\mathrm{1}\right)}\:=\:\underset{\mathrm{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{\mathrm{n}−\mathrm{1}} }{\mathrm{n}}\:−\:\:\mathrm{6}\underset{\mathrm{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{\mathrm{n}−\mathrm{1}} }{\mathrm{6n}+\mathrm{1}} \\ $$$$\Omega\:=\:\mathrm{log}\left(\mathrm{2}\right)\:+\:\left(\pi−\:\mathrm{6}\:+\:\mathrm{2}\sqrt{\mathrm{3}}\mathrm{coth}^{−\mathrm{1}} \left(\sqrt{\mathrm{3}}\right)\right) \\ $$$$\Omega\:\approx\:\mathrm{0}.\mathrm{1157}.. \\ $$