Question

Answer 1

$$\begin{gathered} \mathrm{Let}I=\int e^x\sec x\left(1+\tan x\right)dx \\ =\int e^x\left(\sec x+\sec x\tan x\right)dx \\ \mathrm{Here},f\left(x\right)=\sec x\mathrm{Put}{e}^{x}f\left(x\right)=t \\ \Rightarrow f\text{'}\left(x\right)=\sec x\tan x \\ \mathrm{let}{e}^x\sec x=t \\ \mathrm{Diff}\mathrm{both}\mathrm{sides}\mathrm{w}.\mathrm{r}.\mathrm{t}x \\ {e}^x\sec x+e^x\sec x\tan x=\frac{dt}{dx} \\ \Rightarrow e^x\left(\sec x+\tan x\right)dx=dt \\ \therefore \int e^x\left(\sec x+\sec x\tan x\right)dx=\int dt \\ =t+C \\ =e^x\sec x+C \end{gathered}$$