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7.2 definite integral — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths7.2 definite integral1 Mark
MCQ
$\int_0^\infty {\frac{{{x^3}\,dx}}{{{{({x^2} + 4)}^2}}} = } $
  • A
    $0$
  • ✓
    $\infty $
  • C
    $\frac{1}{2}$
  • D
    None of these
✓

Answer

Correct option: B.
$\infty $
b
(b) $\int_0^\infty {\frac{{{x^3}dx}}{{{{({x^2} + 4)}^2}}} = \frac{1}{2}} \int_0^\infty {\frac{{{x^2}2x\,dx}}{{{{({x^2} + 4)}^2}}}dx} $

$ = \frac{1}{2}\int_0^\infty {\frac{t}{{{{(t + 4)}^2}}}dt} $,

[Putting ${x^2} = t$]

$ = \frac{1}{2}\int_0^\infty {\left[ {\frac{1}{{t + 4}} - \frac{4}{{{{(t + 4)}^2}}}} \right]dt = \frac{1}{2}\left[ {\log (t + 4) + \frac{4}{{t + 4}}} \right]_0^\infty } $

$ = \frac{1}{2}\left[ {\log \infty + 0 - (\log 4 + 1)} \right] = \infty $.

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