MHT CET · Maths · Indefinite Integration
\(\int e^{\tan x}\left(\sec ^2 x+\sec ^3 x \sin x\right) d x=\)
- A \(\tan x \cdot e^{\tan x}+c\)
- B \((1+\tan x) e^{\tan }+c\)
- C \(\sec x \cdot e^{\tan x}+c\)
- D \(\mathrm{e}^{\tan x+\tan x}+c\)
Answer & Solution
Correct Answer
(A) \(\tan x \cdot e^{\tan x}+c\)
Step-by-step Solution
Detailed explanation
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 \(\tan \mathrm{x}=\mathrm{t} \Rightarrow \sec ^2 \mathrm{xdx}=\mathrm{dt}\)
\(
\therefore I=\int \mathrm{e}^{\mathrm{t}}(1+\mathrm{t}) \mathrm{dt}=\mathrm{e}^{\mathrm{t}}(\mathrm{t})+\mathrm{c}=\mathrm{e}^{\tan \mathrm{x}}(\tan \mathrm{x})+\mathrm{c}
\)
\(
I=\int e^{\tan x}\left(\sec ^2 x+\sec ^3 x \sin x\right) d x=\) \(\int e^{\tan x}\left(\sec ^2 x\right)+(1+\tan x) d x
\)
Put \(\tan \mathrm{x}=\mathrm{t} \Rightarrow \sec ^2 \mathrm{xdx}=\mathrm{dt}\)
\(
\therefore I=\int \mathrm{e}^{\mathrm{t}}(1+\mathrm{t}) \mathrm{dt}=\mathrm{e}^{\mathrm{t}}(\mathrm{t})+\mathrm{c}=\mathrm{e}^{\tan \mathrm{x}}(\tan \mathrm{x})+\mathrm{c}
\)
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