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7.1 inderfinite integral — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths7.1 inderfinite integral1 Mark
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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