Question

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

• A. $\frac{x^{52}}{52}\left({\tan }^{-1}x{\cot }^{-1}x\right)+c$
• B. $\frac{x^{52}}{51}\left({\tan }^{-1}x{\cot }^{-1}x\right)+c$
• C. $-\frac{\pi x^{52}}{104}+\frac{\pi }{2}+c$
• D. $\frac{x^{52}}{52}+\frac{\pi }{2}+c$

Answer 1

The correct option is A $\frac{x^{52}}{52}\left({\tan }^{-1}x{\cot }^{-1}x\right)+c$

Let $I=\int x^{51}\left({\tan }^{-1}x{\cot }^{-1}x\right)dx$

Now,

${\tan }^{-1}x{\cot }^{-1}x=\frac{1}{{\cot }^{-1}x}.{\cot }^{-1}x=1$

$I=\int x^{51}dx$

$I=\frac{x^{51+1}}{51+1}+c$

$I=\frac{x^{52}}{52}+c$

$I=\frac{x^{52}}{52}\left(1\right)+c$

$I=\frac{x^{52}}{52}\left({\tan }^{-1}x{\cot }^{-1}x\right)+c$