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प्रश्न
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उत्तर
\[\int e^x \left( \tan x + 1 \right) \text{ sec x dx} = \int e^x \left( \tan x\sec x + \sec x \right) dx\]
\[ = \int e^x \left( \sec x + \tan x\sec x \right) dx\]
\[\text{ Consider}, f\left( x \right) = \sec x,\text{ then f}^{ ' } \left( x \right) = \tan x\sec x\]
\[\text{ Thus , the given integrand is of the form e}^x \left[ f\left( x \right) + f^{ '} \left( x \right) \right] . \]
\[\text{ Therefore,} \int e^x \left( \tan x + 1 \right) \text{ sec x dx} = \sec x \text{ e}^x + C\]
\[\text{ Hence,} f\left( x \right) = \sec x .\]
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