Question

Assertion :Statement 1:$F\left(x\right)={\int }_1^x\frac{\log t}{1+t+t^2}dt$ then $F\left(x\right)=-F\left(1/x\right)$ Reason: Statement 2: If $F\left(x\right)={\int }_1^x\frac{\log t}{t+1}dt$ then $F\left(x\right)+F\left(1/x\right)=\left(1/2\right)(\log x{)}^2$

• 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 D Both Assertion and Reason are incorrect

$F\left(1/x\right)={\int }_1^{1/x}\frac{\log t}{1+t+t^2}dt$

$={\int }_1^x\frac{\log \left(1/u\right)}{1+\frac{1}{u}+\frac{1}{u^2}}(-\frac{du}{u^2}),(u=\frac{1}{t})$

$={\int }_1^x\frac{\log u}{u^2+u+1}du=F\left(x\right)$

So statement 1 is not true

If $F\left(x\right)={\int }_1^x\frac{\log t}{t+1}dt$, then

$F\left(x\right)+F\left(1/x\right)={\int }_1^x\frac{\log t}{t+1}dt+{\int }_1^{1/x}\frac{\log t}{t+1}dt$

$={\int }_1^x\frac{\log t}{t+1}dt+{\int }_1^x\frac{\log \left(1/u\right)}{1+1/u}(-\frac{du}{u^2})$

$={\int }_1^x\log t(\frac{1}{t+1}+\frac{1}{t^2+t})dt$

$={\int }_1^x\frac{\log t}{t+1}(1+\frac{1}{t})dt$

$={\int }_1^x\frac{\log t}{t}dt=-\frac{(\log x{)}^2}{2}$

Hence both statements are incorrect.