What is the integration of \[ \int_0^1 {\frac{{dx}}{{{e^x} + {e^{ – x}}}}} \] ~~~~~[NCERT Exemp. Q. 29,Page 165 ]

Let \[ I = \int_0^1 {\frac{{dx}}{{{e^x} + {e^{ – x}}}}} = \int_0^1 {\frac{{{e^x}}}{{1 + {e^{2x}}}}} dx\]

Let’s put \[ {e^x} = t\]

\[ \Rightarrow \] \[{e^x}dx = dt\]

therefore,\[ I = \int_1^e {\frac{{dt}}{{1 + {t^2}}}} = \left[ {{{\tan }^{ – 1}}t} \right]_1^e\]

\[ = {\tan ^{ – 1}}e – {\tan ^{ – 1}}1\]

\[ = {\tan ^{ – 1}}e – \frac{\pi }{4}\]

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