MCQ
$\int_0^\infty {\frac{{x\,dx}}{{(1 + x)(1 + {x^2})}}} = $
- ✓$\frac{\pi }{4}$
- B$\frac{\pi }{3}$
- C$\frac{\pi }{6}$
- DNone of these
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}} } $.
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$[A]$ $e^x-\int_0^x f(t) \sin t d t$ $[B]$ $x^9-f(x)$ $[C]$ $f(x)+\int_0^{\pi / 2} f(t) \sin t d t$
$[D]$ $x-\int_0^{\frac{\pi}{2}-x} f(t) \cos t d t$