Question

$\int x ^{51}\left(\tan ^{-1} x +\cot ^{-1} x \right) d x$

• A. $\frac{x^{52}}{52}\left(\tan ^{-1} x+\cot ^{-1} x\right)+c$
• B. $\frac{x^{52}}{52}\left(\tan ^{-1} x-\cot ^{-1} x\right)+c$
• C. $\frac{\pi x^{52}}{104}+\frac{\pi}{2}+c$
• D. $\frac{x^{52}}{52}+\frac{\pi}{2}+c$

Answer 1

Answer: (A)

$\int x^{51}\left(\tan ^{-1} x+\cot ^{-1} x\right) d x$
$=\int x ^{51} \cdot \frac{\pi}{2} dx\,\, \left\{\because \tan ^{-1} x +\cot ^{-1} x =\frac{\pi}{2}\right\}$
$=\frac{\pi x ^{52}}{104}+ c =\frac{ x ^{52}}{52}\left(\tan ^{-1} x +\cot ^{-1} x \right)+ c .$