A) \[\frac{{{x}^{52}}}{52}({{\tan }^{-1}}x+{{\cot }^{-1}}x)+c\]
B) \[\frac{{{x}^{52}}}{52}({{\tan }^{-1}}x-{{\cot }^{-1}}x)+c\]
C) \[\frac{\pi {{x}^{52}}}{104}+\frac{\pi }{2}+c\]
D) \[\frac{{{x}^{52}}}{52}+\frac{\pi }{2}+c\]
Correct Answer: A
Solution :
\[\int_{{}}^{{}}{{{x}^{51}}({{\tan }^{-1}}x+{{\cot }^{-1}}x)}\,dx=\int_{{}}^{{}}{{{x}^{51}}.\frac{\pi }{2}dx}\] \[\left\{ \because \ {{\tan }^{-1}}x+{{\cot }^{-1}}x=\frac{\pi }{2} \right\}\] \[=\frac{\pi \,{{x}^{52}}}{104}+c=\frac{{{x}^{52}}}{52}({{\tan }^{-1}}x+{{\cot }^{-1}}x)+c\].
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