Q96 / 150 · MHT CET 2021 (21 Sep Shift 1) · Mathematics

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

  • Correct answer
    A
    $\tan x \cdot e^{\tan x}+c$
  • B
    $(1+\tan x) e^{\tan }+c$
  • C
    $\sec x \cdot e^{\tan x}+c$
  • D
    $\mathrm{e}^{\tan x+\tan x}+c$

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 $\tan \mathrm{x}=\mathrm{t} \Rightarrow \sec ^2 \mathrm{xdx}=\mathrm{dt}$ $$ \therefore I=\int \mathrm{e}^{\mathrm{t}}(1+\mathrm{t}) \mathrm{dt}=\mathrm{e}^{\mathrm{t}}(\mathrm{t})+\mathrm{c}=\mathrm{e}^{\tan \mathrm{x}}(\tan \mathrm{x})+\mathrm{c} $$

Hence, the correct answer is option: 'A'.

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