Question

$\int^{3}_{-3} \cot^{-1} x\, dx = $

• A. $6 \pi $
• B. $3 \pi $
• C. 3
• D. 0

Answer 1

Answer: (B)

$\int\limits_{-3}^{3} cot^{-1} x dx=\int\limits_{-3}^{3}cot^{-1} x. 1dx$
$=\left[cot^{-1}x.x\right]_{-3}^{3}-\int\limits_{-3}^{3}x \left(\frac{1}{1+x^{2}}\right)dx$
$=3 cot^{-1}3-\left(-3\right)cot^{-1}\left(-3\right)+\int\limits_{-3}^{3} \frac{x}{1+x^{2}} dx$
$=3 cot^{-1}3+3\left(\pi-cot^{-1}3\right)+\frac{1}{2}\left[log\left[1+x^{2}\right]\right]_{-3}^{3}$
$=3 cot^{-1}3+3\pi-3cot^{-1}3+\frac{1}{2}\left[log 10-log 10\right]$
$ = 3 \pi $