Question

${\int }_{-3}^3{\cot }^{-1}xdx=$

• A. $6\pi$
• B. $3\pi$
• C. 3
• D. 0

Answer 1

Answer: (B)

$\underset{-3}{\overset{3}{\int }}co{t}^{-1}xdx=\underset{-3}{\overset{3}{\int }}co{t}^{-1}x.1dx$
$={\left[co{t}^{-1}x.x\right]}_{-3}^3-\underset{-3}{\overset{3}{\int }}x\left(\frac{1}{1+x^2}\right)dx$
$=3co{t}^{-1}3-\left(-3\right)co{t}^{-1}\left(-3\right)+\underset{-3}{\overset{3}{\int }}\frac{x}{1+x^2}dx$
$=3co{t}^{-1}3+3\left(\pi -co{t}^{-1}3\right)+\frac{1}{2}{\left[log\left[1+x^2\right]\right]}_{-3}^3$
$=3co{t}^{-1}3+3\pi -3co{t}^{-1}3+\frac{1}{2}\left[log10-log10\right]$
$=3\pi$