Integral Calculus Question 260
Question: $ \int_{{}}^{{}}{x^{51}({{\tan }^{-1}}x+{{\cot }^{-1}}x)\ dx=} $
[MP PET 1991]
Options:
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 $
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Answer:
Correct Answer: A
Solution:
$ \int_{{}}^{{}}{x^{51}({{\tan }^{-1}}x+{{\cot }^{-1}}x)},dx=\int_{{}}^{{}}{x^{51}.\frac{\pi }{2}dx} $ $ { \because \ {{\tan }^{-1}}x+{{\cot }^{-1}}x=\frac{\pi }{2} } $ $ =\frac{\pi ,x^{52}}{104}+c=\frac{x^{52}}{52}({{\tan }^{-1}}x+{{\cot }^{-1}}x)+c $ .
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