Question

\[If \int e^x \left( \tan x + 1 \right)\text{ sec  x  dx } = e^x f\left( x \right) + C, \text{ then  write  the value  of  f}\left( x \right) .\]

 

 

Answer 1

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