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Question Number 33494 by mondodotto@gmail.com last updated on 17/Apr/18

$$\int \frac{{\mathbf{e}}^{\mathit{x}}}{{\mathbf{sin}}^2\mathit{x}}\mathit{d}\mathit{x}$$

Commented by math khazana by abdo last updated on 18/Apr/18

$$\int \frac{e^x}{{sin}^{2}x}dx=\int \frac{e^x}{\frac{1-cos\left(2x\right)}{2}}dx$$$$=2\int \frac{e^x}{1-cos\left(2x\right)}dx$$$${=}_{2x=t}2\int \frac{e^{\frac{t}{2}}}{1-cost}\frac{dt}{2}$$

Commented by math khazana by abdo last updated on 18/Apr/18

$$\int \frac{e^x}{{sin}^{2}x}dx=\int e^{\frac{t}{2}}\left(\sum _{n=0}^{\mathrm{\infty }}{cos}^{n}t\right)dt$$$$=\sum _{n=0}^{\mathrm{\infty }}\int e^{\frac{t}{2}}{cos}^{n}tdt=\sum _{n=0}^{\mathrm{\infty }}{A}_{n}with$$$$A_n=\int e^{\frac{t}{2}}{cos}^{n}tdtintegralcalculablebyrecurence$$

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