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STD 12 - 7.2 definite integral — Mathematics STD 12 Science — Question

CBSE BoardEnglish MediumSTD 12 ScienceMathematicsSTD 12 - 7.2 definite integral1 Mark
MCQ
$\int_0^\infty {\frac{{x\,dx}}{{(1 + x)(1 + {x^2})}}} = $
  • ✓
    $\frac{\pi }{4}$
  • B
    $\frac{\pi }{3}$
  • C
    $\frac{\pi }{6}$
  • D
    None of these
✓

Answer

Correct option: A.
$\frac{\pi }{4}$
a
(a) $I = \int_0^\infty {\frac{{xdx}}{{(1 + x)(1 + {x^2})}}} $

Put $x = \tan \theta $, we get

$I = \int_0^{\pi /2} {\frac{{\tan \theta }}{{1 + \tan \theta }}d\theta = \int_0^{\pi /2} {\frac{{\sin \theta }}{{\cos \theta + \sin \theta }}d\theta = \frac{\pi }{4}} } $.

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