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Question Number 198901 by sonukgindia last updated on 25/Oct/23

Answered by witcher3 last updated on 25/Oct/23

$$\int \frac{{(\cos \left(\frac{\mathrm{x}}{2}\right)-\sin \left(\frac{\mathrm{x}}{2}\right))}^2}{6\sin \left(\frac{\mathrm{x}}{2}\right)\cos \left(\frac{\mathrm{x}}{2}\right)+{10\cos }^2\left(\frac{\mathrm{x}}{2}\right){\mathrm{e}}^{\mathrm{x}}}\mathrm{dx}=\mathrm{g}\left(\mathrm{x}\right)$$$$=\int \frac{{(1-\mathrm{tg}\left(\frac{\mathrm{x}}{2}\right))}^2}{6\mathrm{tg}\left(\frac{\mathrm{x}}{2}\right)+{10\mathrm{e}}^{\mathrm{x}}}\mathrm{dx}$$$$=\frac{1}{10}\int \frac{({\mathrm{tg}}^2\left(\frac{\mathrm{x}}{2}\right)-2\mathrm{tg}\left(\frac{\mathrm{x}}{2}\right)+1){\mathrm{e}}^{-\mathrm{x}}}{\frac{3}{5}\mathrm{tg}\left(\frac{\mathrm{x}}{2}\right){\mathrm{e}}^{-\mathrm{x}}+1}$$$$\mathrm{f}\left(\mathrm{x}\right)=\frac{3}{5}\mathrm{tg}\left(\frac{\mathrm{x}}{2}\right){\mathrm{e}}^{-\mathrm{x}}+1$$$${\mathrm{f}}^{\prime }\left(\mathrm{x}\right)=\frac{3}{5}(\frac{1}{2}(1+{\mathrm{tg}}^2\left(\frac{\mathrm{x}}{2}\right))-\mathrm{tg}\left(\frac{\mathrm{x}}{2}\right)){\mathrm{e}}^{-\mathrm{x}}$$$$=\frac{3}{10}({\mathrm{tg}}^2\left(\frac{\mathrm{x}}{2}\right)+1-2\mathrm{tg}\left(\frac{\mathrm{x}}{2}\right)){\mathrm{e}}^{-\mathrm{x}}$$$$\mathrm{g}\left(\mathrm{x}\right)=\frac{1}{10}\int .\frac{10}{3}.\frac{{\mathrm{f}}^{\prime }}{\mathrm{f}}$$$$\frac{1}{3}\ln (\mid \frac{3}{5}\mathrm{tg}\left(\frac{\mathrm{x}}{2}\right){\mathrm{e}}^{-\mathrm{x}}+1\mid )+\mathrm{c},\mathrm{c}\in \mathbb{R}$$$$$$

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