$\int e^{\tan x}\left(\sec ^2 x+\sec ^3 x \sin x\right) d x=$

$\int e^{\tan x}\left(\sec ^2 x+\sec ^3 x \sin x\right) d x=$
  1. $\tan x \cdot e^{\tan x}+c$
  2. $(1+\tan x) e^{\tan }+c$
  3. $\sec x \cdot e^{\tan x}+c$
  4. $\mathrm{e}^{\tan x+\tan x}+c$

Solution

Let $I=\int e^{\tan x}\left(\sec ^2 x+\sec ^3 x \sin x\right) d x=\int e^{\tan x}\left(\sec ^2 x\right)+(1+\tan x) d x$ Put $\tan \mathrm{x}=\mathrm{t} \Rightarrow \sec ^2 \mathrm{xdx}=\mathrm{dt}$ $\therefore I=\int \mathrm{e}^{\mathrm{t}}(1+\mathrm{t}) \mathrm{dt}=\mathrm{e}^{\mathrm{t}}(\mathrm{t})+\mathrm{c}=\mathrm{e}^{\tan \mathrm{x}}(\tan \mathrm{x})+\mathrm{c}$

Asked in: MHT CET 2021 (21 Sep Shift 1)

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