Mathematics
##### ASSERTION

If $\int \sec \left ( \log x \right )\left ( 1+\tan \left ( \log x \right ) \right )dx=f\left ( x \right )+C$ then $f\left ( x \right )=x\sec \left ( \log x \right )$

##### REASON

$\int \left \{ x{f}'\left ( x \right )+f\left ( x \right ) \right \}dx=xf\left ( x \right )+C$

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

##### SOLUTION
Given $\int \sec \left ( \log x \right )\left ( 1+\tan \left ( \log x \right ) \right )dx=f\left ( x \right )+C$        ....(1)
Consider, $I= \int \sec \left ( \log x \right )\left ( 1+\tan \left ( \log x \right ) \right )dx$
Putting $\log x=t$
$dx=e^{t}dt$
$I=\int e^{t}\left ( \sec t+\sec t\tan t \right )dt$
$=e^{t}\sec t+C$              ($\because \int e^x(f(x)+f'(x))=e^f(x)+C$)
$=x\sec \left ( \log x \right )+C$
So, by (1), we get
$f\left ( x \right )=x\sec \left ( \log x \right )$
$\therefore$   Assertion (A) is true.

Reason(R): $\int \left \{ f\left ( x \right )+x{f}'\left ( x \right ) \right \}dx$
$=\int f\left ( x \right )dx+\int x{f}'\left ( x \right )dx$
$=f\left ( x \right )+xf\left ( x \right )-\int 1.f\left ( x \right )dx$
$=xf\left ( x \right )+C$
$\therefore$   Reason (R) is also true but not the proper explanation for Assertion (A).

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Assertion & Reason Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

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