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Question Number 123255 by mnjuly1970 last updated on 24/Nov/20

          ∗∗∗  nice  calculus ∗∗∗         evaluate ::            Φ=∫_0 ^(π/2) log^3 (tan(x))dx =?

$$\:\:\:\:\:\:\:\:\:\:\ast\ast\ast\:\:{nice}\:\:{calculus}\:\ast\ast\ast \\ $$$$\:\:\:\:\:\:\:{evaluate}\::: \\ $$$$\:\:\:\:\:\:\:\:\:\:\Phi=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {log}^{\mathrm{3}} \left({tan}\left({x}\right)\right){dx}\:=? \\ $$

Answered by Dwaipayan Shikari last updated on 24/Nov/20

∫_0 ^(π/2) log^3 (tanx)dx=I  =∫_0 ^(π/2) log^3 (cotx)=∫_0 ^(π/2) −log^3 (tanx)=I  2I=0⇒I=0

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {log}^{\mathrm{3}} \left({tanx}\right){dx}={I} \\ $$$$=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {log}^{\mathrm{3}} \left({cotx}\right)=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} −{log}^{\mathrm{3}} \left({tanx}\right)={I} \\ $$$$\mathrm{2}{I}=\mathrm{0}\Rightarrow{I}=\mathrm{0} \\ $$

Commented by mnjuly1970 last updated on 24/Nov/20

   thank you...

$$\:\:\:{thank}\:{you}... \\ $$

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