\(\rm \int\frac{e^{tan^{-1}x}}{1+x^2}dx=?\)

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AAI ATC Junior Executive 21 Feb 2023 Shift 1 Official Paper
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  1. \(\rm e^{\tan ^{-1}x}+c\)
  2. \(\rm e^{-\tan ^{-1}x}+c\)
  3. \(\rm e^{-\tan ^{-1}x}\)
  4. \(\rm e^{\tan x}+c\)

Answer (Detailed Solution Below)

Option 1 : \(\rm e^{\tan ^{-1}x}+c\)
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Detailed Solution

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Explanation -

We have \(I =\rm \int\frac{e^{tan^{-1}x}}{1+x^2}dx\)

Let \(​​​​​​​​tan^{-1}x= t \implies \frac{1}{1+x^2} dx =dt\)

Now we get -

\(I= \int e^t dt= e^t +C\)

\(I= e^{tan^{-1}x} +C\)

Hence the option (1) is correct.

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