Question Number 123255 by mnjuly1970 last updated on 24/Nov/20
$$\:\:\:\:\:\:\:\:\:\:\ast\ast\ast\:\:{nice}\:\:{calculus}\:\ast\ast\ast \\ $$$$\:\:\:\:\:\:\:{evaluate}\::: \\ $$$$\:\:\:\:\:\:\:\:\:\:\Phi=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {log}^{\mathrm{3}} \left({tan}\left({x}\right)\right){dx}\:=? \\ $$
Answered by Dwaipayan Shikari last updated on 24/Nov/20
$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {log}^{\mathrm{3}} \left({tanx}\right){dx}={I} \\ $$$$=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {log}^{\mathrm{3}} \left({cotx}\right)=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} −{log}^{\mathrm{3}} \left({tanx}\right)={I} \\ $$$$\mathrm{2}{I}=\mathrm{0}\Rightarrow{I}=\mathrm{0} \\ $$
Commented by mnjuly1970 last updated on 24/Nov/20
$$\:\:\:{thank}\:{you}… \\ $$