Question

Assertion :${\int }_{-1}^3\{{\tan }^{-1}\left(\frac{x}{1+x^2}\right)+{\tan }^{-1}\left(\frac{x^2+1}{x}\right)\}dx=2\pi$ Reason: ${\tan }^{-1}x+{\cot }^{-1}x=\frac{\pi }{2}$ and ${\tan }^{-1}t={\cot }^{-1}\frac{1}{t}$

• 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 correct

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

${\int }_{-1}^3\{{\tan }^{-1}\left(\frac{x}{1+x^2}\right)+{\tan }^{-1}\left(\frac{1+x^2}{x}\right)\}dx$
$={\int }_{-1}^3\{{\tan }^{-1}\left(\frac{x}{1+x^2}\right)+{\cot }^{-1}\left(\frac{x}{1+x^2}\right)\}dx$
$=\frac{\pi }{2}{\int }_{-1}^{3}dx$ as Reason (R) is correct $[\because {\tan }^{-1}x+{\cot }^{-1}x=\frac{\pi }{2}]$
$=\frac{\pi }{2}(3+1)=2\pi$

$\therefore {\int }_{-1}^3\{{\tan }^{-1}\left(\frac{x}{1+x^2}\right)+{\tan }^{-1}\left(\frac{1+x^2}{x}\right)\}dx=2\pi$

Hence, both assertion and reason are correct and reason is the correct explanation for assertion.