Question

Assertion :If $\int \sec \left(\log x\right)(1+\tan \left(\log x\right))dx=f\left(x\right)+C$ then $f\left(x\right)=x\sec \left(\log x\right)$ Reason: $\int \{x{f}^{\prime }\left(x\right)+f\left(x\right)\}dx=xf\left(x\right)+C$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

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

Given $\int \sec \left(\log x\right)(1+\tan \left(\log x\right))dx=f\left(x\right)+C$ ....(1)
Consider, $I=\int \sec \left(\log x\right)(1+\tan \left(\log x\right))dx$
Putting $\log x=t$
$dx=e^{t}dt$
$I=\int e^t(\sec t+\sec t\tan t)dt$
$=e^t\sec t+C$ ($\because \int e^x\left(f\right(x)+f^{\prime }(x\left)\right)=e^f\left(x\right)+C$)
$=x\sec \left(\log x\right)+C$
So, by (1), we get
$f\left(x\right)=x\sec \left(\log x\right)$
$\therefore$ Assertion (A) is true.
Reason(R): $\int \{f\left(x\right)+x{f}^{\prime }\left(x\right)\}dx$
$=\int f\left(x\right)dx+\int x{f}^{\prime }\left(x\right)dx$
$=f\left(x\right)+xf\left(x\right)-\int 1.f\left(x\right)dx$
$=xf\left(x\right)+C$
$\therefore$ Reason (R) is also true but not the proper explanation for Assertion (A).