Question

Assertion :If $\int \frac{1}{f\left(x\right)}dx=\log {\left(f\left(x\right)\right)}^2+C$, then $f\left(x\right)=\frac{x}{2}$ Reason: When $f\left(x\right)=\frac{x}{2}$ then $\int \frac{1}{f\left(x\right)}dx=\int \frac{2}{x}dx=2\log \left|x\right|+C$

• 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

Assertion
$\int \frac{1}{f\left(x\right)}dx=\log {\left(f\left(x\right)\right)}^2+C$
$\int \frac{1}{f\left(x\right)}dx=2\log \left|f\left(x\right)\right|+C$
$\therefore \frac{d}{dx}(\int \frac{1}{f\left(x\right)}dx)=2\frac{d}{dx}\left(\log \left|f\left(x\right)\right|\right)+\frac{d}{dx}\left(C\right)$
$\frac{1}{f\left(x\right)}=\frac{2}{f\left(x\right)}.f^{\prime }\left(x\right)$
$\Rightarrow f^{\prime }\left(x\right)=\frac{1}{2}$
$\Rightarrow f\left(x\right)=\frac{x}{2}+C$
Reason : $f\left(x\right)=\frac{x}{2}$
$\frac{1}{f\left(x\right)}=\frac{2}{x}$
$\Rightarrow \int \frac{1}{f\left(x\right)}dx=\int \frac{2}{x}dx$
$\Rightarrow \int \frac{1}{f\left(x\right)}dx=2\log \left|x\right|+C$
Reason is correct but not the explanation for assertion