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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{{\log \,(1 + {x^2})}}{{1 + {x^2}}}} \,dx = $
  • A
    $\pi \log \frac{1}{2}$
  • ✓
    $\pi \log 2$
  • C
    $2\pi \log \frac{1}{2}$
  • D
    $2\pi \log 2$
✓

Answer

Correct option: B.
$\pi \log 2$
b
(b) Let $I = \int_0^\infty {\frac{{\log (1 + {x^2})}}{{1 + {x^2}}}\,\,dx} $

Put $x = \tan \theta \Rightarrow dx = {\sec ^2}\theta \,d\theta ,$

$\therefore $ $I = \int_0^{\pi /2} {\log {{(\sec \theta )}^2}d\theta = 2\int_0^{\pi /2} {\log \sec \theta \,\,d\theta } } $

$ = - 2\int_0^{\pi /2} {\log \cos \theta \,\,d\theta = - 2.\,\,\frac{\pi }{2}\log \frac{1}{2}} $

$ = - \pi \log \frac{1}{2} = \pi \log 2$.

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