_ ^ x^ 51 ( ^ - 1 x + ^ - 1 x) ;dx = - Vidyadip

MCQ

$\int_{}^{} {{x^{51}}({{\tan }^{ - 1}}x + {{\cot }^{ - 1}}x)\;dx = } $

✓
$\frac{{{x^{52}}}}{{52}}({\tan ^{ - 1}}x + {\cot ^{ - 1}}x) + c$
B
$\frac{{{x^{52}}}}{{52}}({\tan ^{ - 1}}x - {\cot ^{ - 1}}x) + c$
C
$\frac{{\pi {x^{52}}}}{{104}} + \frac{\pi }{2} + c$
D
$\frac{{{x^{52}}}}{{52}} + \frac{\pi }{2} + c$

✓

Answer

Correct option: A.

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

a
(a) $\int_{}^{} {{x^{51}}({{\tan }^{ - 1}}x + {{\cot }^{ - 1}}x)} \,dx = \int_{}^{} {{x^{51}}.\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}}({\tan ^{ - 1}}x + {\cot ^{ - 1}}x) + c$.

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