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

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

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
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

#### Realted Questions

Q1 Subjective Hard
Let $p\left(x\right)$ be the fifth degree polynomial such that $p\left(x\right) +1$  is divisible by $\left(x-1\right)$ and $p\left(x\right) - 1$ is divisible by $\left(x+1\right)$.Then find the value of $\int _{ -10 }^{ 10 }{ p(x)dx }$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Evaluate the following integral as the limit of sum:
$\displaystyle\int^4_2(x^2-3x+2)dx$.

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Hard
Evaluate : $\displaystyle \int_0^{\pi} \dfrac{x \sin x}{1 + \sin x} dx$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Evaluate:
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Evaluate $\int \dfrac{e^x-e^{-x}}{e^x+e^{-x}}dx$