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Question Number 89946 by jagoll last updated on 20/Apr/20

$$\int \underset{-\frac{\pi }{2}}{\stackrel{\frac{\pi }{2}}{}}\frac{\mathrm{dx}}{1+{\mathrm{e}}^{\sin \mathrm{x}}}$$

Answered by john santu last updated on 20/Apr/20

$$I=\underset{\frac{-\pi }{2}}{\stackrel{\frac{\pi }{2}}{\int }}\left(\frac{dx}{1+e^{\sin x}}\right)$$$$replacexby-x$$$$I=\underset{\frac{\pi }{2}}{\stackrel{-\frac{\pi }{2}}{\int }}\left(\frac{-dx}{1+e^{-\sin x}}\right)=\underset{-\frac{\pi }{2}}{\stackrel{\frac{\pi }{2}}{\int }}\left(\frac{dx}{1+e^{-\sin x}}\right)$$$$I+I=\underset{-\frac{\pi }{2}}{\stackrel{\frac{\pi }{2}}{\int }}\frac{1+e^{\sin x}}{1+e^{\sin x}}dx$$$$2I=\underset{-\frac{\pi }{2}}{\stackrel{\frac{\pi }{2}}{\int }}1dx=\pi$$$$I=\frac{\pi }{2}$$

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