Mathematics
ASSERTION

$$\displaystyle \int e^{\tan^{-1}x}\left ( \frac{1+x+x^{2}}{1+x^{2}} \right )dx=x\tan^{-1}x+C$$

REASON

$$\int e^{t}\left ( f\left ( t \right )+{f}'\left ( t \right ) \right )dt=e^{t}f\left ( t \right )+C$$


ANSWER

Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion


SOLUTION
Reason is true (theory)
$$\displaystyle I=\int  e^{ \tan ^{ -1 }  }x\left( \frac { 1+x+x^{ 2 } }{ 1+x^{ 2 } }  \right) dx\\ \tan ^{ -1 } x=t\Rightarrow x=\tan  t\Rightarrow 1+x^{ 2 }=\sec ^{ 2 } t\\ I=\int  e^{ t }\left( \tan  t+\sec ^{ 2 } t \right) dt=e^{ t }\tan  t+C=xe^{ \tan ^{ -1 } x }+C$$
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Assertion & Reason Medium Published on 17th 09, 2020
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