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Question Number 188380 by Shlock last updated on 28/Feb/23

Answered by SEKRET last updated on 28/Feb/23

$$\mathbf{f}\left(\mathbf{t}\right)={\int }_0^{\mathit{\pi }}\frac{\mathbf{ln}(1+\mathbf{t}\cdot \mathbf{sin}\left(\mathbf{a}\right)\cdot \mathbf{cos}\left(\mathbf{x}\right))}{\mathbf{cos}\left(\mathbf{x}\right)}\mathbf{dx}$$$$\mathbf{f}{}^{\prime }\left(\mathbf{t}\right)={\int }_0^{\mathit{\pi }}\frac{\mathbf{sin}\left(\mathbf{a}\right)\cdot \mathbf{cos}\left(\mathbf{x}\right)}{\mathbf{cos}\left(\mathbf{x}\right)\cdot (1+\mathbf{t}\cdot \mathbf{sin}\left(\mathbf{a}\right)\cdot \mathbf{cos}\left(\mathbf{x}\right))}\mathbf{dx}$$$$\mathbf{f}{}^{\prime }\left(\mathbf{t}\right)=\mathbf{sin}\left(\mathbf{a}\right)\cdot {\int }_0^{\mathit{\pi }}\frac{1}{1+\mathit{t}\cdot \mathbf{sin}\left(\mathbf{a}\right)\cdot \mathbf{cos}\left(\mathbf{x}\right)}\mathbf{dx}$$$$\mathbf{sin}\left(\mathbf{a}\right)\cdot {\left[\frac{2\mathbf{arctg}\left(\frac{\mathbf{tg}\left(\frac{\mathbf{x}}{2}\right)\cdot (\mathbf{t}\cdot \mathbf{sin}\left(\mathbf{a}\right)-1)}{\sqrt{1-{\mathbf{t}}^2{\mathbf{sin}}^2\left(\mathbf{a}\right)}}\right)}{\sqrt{1-{\mathbf{t}}^2\cdot {\mathbf{sin}}^2\left(\mathbf{a}\right)}}\right]}_0^{\mathit{\pi }}$$$$=\mathbf{sin}\left(\mathbf{a}\right)\cdot \left(\frac{\mathit{\pi }}{\sqrt{1-{\mathbf{t}}^2\cdot {\mathbf{sin}}^2\mathbf{a}}}\right)$$$$\mathbf{f}\left(\mathbf{t}\right)=\mathit{\pi }\cdot \mathbf{sin}\left(\mathbf{a}\right)\cdot \int \frac{1}{\sqrt{1-{\mathbf{t}}^2{\mathbf{sin}}^2\mathbf{a}}}\mathbf{dt}$$$$\mathbf{f}\left(\mathbf{t}\right)=\mathit{\pi }\cdot \mathbf{sin}\left(\mathbf{a}\right)\cdot \frac{\mathbf{arcsin}(\mathbf{t}\cdot \mathbf{sin}\left(\mathbf{a}\right))}{\mathbf{sin}\left(\mathbf{a}\right)}$$$$\mathbf{t}=1\mathbf{I}=\mathit{\pi }\cdot \mathbf{a}$$$${\int }_0^{\mathit{\pi }}\frac{\mathbf{ln}(1+\mathbf{sin}\left(\mathbf{a}\right)\cdot \mathbf{cos}\left(\mathbf{x}\right))}{\mathbf{cos}\left(\mathbf{x}\right)}\mathbf{dx}=\mathit{\pi }\cdot \mathbf{a}$$$$\mathit{A}\mathit{B}\mathit{D}\mathit{U}\mathit{L}\mathit{A}\mathit{Z}\mathit{I}\mathit{Z}\mathit{A}\mathit{B}\mathit{D}\mathit{U}\mathit{V}\mathit{A}\mathit{L}\mathit{I}\mathit{Y}\mathit{E}\mathit{V}$$$$$$

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