Question

$\mathrm{If}\int {\mathrm{e}}^x\left(\tan x+1\right)\sec x\mathrm{d}x={\mathrm{e}}^{x}f\left(x\right)+C,\mathrm{then}\mathrm{write}\mathrm{the}\mathrm{value}\mathrm{of}f\left(x\right).$

Answer 1

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