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Question Number 184497 by Noorzai last updated on 07/Jan/23

Answered by SEKRET last updated on 07/Jan/23

$${\mathbf{x}}^{{\mathbf{x}}^{{\mathbf{x}}^{....}}}=\mathbf{y}{\mathbf{x}}^{\mathbf{y}}=\mathbf{y}$$$$\mathbf{x}={\mathbf{y}}^{\frac{1}{\mathbf{y}}}\mathbf{dx}=-{\mathbf{y}}^{\frac{1}{\mathbf{y}}-2}\cdot (\mathbf{ln}\left(\mathbf{y}\right)-1)\mathbf{dy}$$$$\int \mathbf{y}\cdot (-1)\cdot {\mathbf{y}}^{\frac{1}{\mathbf{y}}-2}(\mathbf{lny}-1)\mathbf{dy}=$$$$-1\cdot \int {\mathbf{y}}^{\frac{1}{\mathbf{y}}-1}\cdot (\mathbf{ln}\left(\mathbf{y}\right)-1)\mathbf{dy}=$$$$=\int {\mathbf{y}}^{\frac{1}{\mathbf{y}}-1}\mathbf{dy}-\int {\mathbf{y}}^{\frac{1}{\mathbf{y}}-1}\cdot \mathbf{ln}\left(\mathbf{y}\right)\mathbf{dy}$$$$......$$$$$$$$$$

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