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RAJASTHAN ­ PET Rajasthan PET Solved Paper-2005

  • question_answer
    If \[\int_{0}^{\infty }{\frac{{{x}^{2}}dx}{({{x}^{2}}+{{a}^{2}})({{x}^{2}}+{{b}^{2}})({{x}^{2}}+{{c}^{2}})}}\] \[=\frac{\pi }{2(a+b)(b+c)(c+a)},\]then the value of \[\int_{0}^{\infty }{\frac{dx}{({{x}^{2}}+4)({{x}^{2}}+9)}}\]is

    A)  \[\frac{\pi }{20}\]

    B)  \[\frac{\pi }{40}\]

    C)  \[\frac{\pi }{60}\]

    D)  \[\frac{\pi }{80}\]

    Correct Answer: C

    Solution :

     Given, \[\int_{0}^{\infty }{\frac{{{x}^{2}}dx}{({{x}^{2}}+{{a}^{2}})({{x}^{2}}+{{b}^{2}})({{x}^{2}}+{{c}^{2}})}}\] \[=\frac{\pi }{2(a+b)(b+c)(c+a)}\] Put\[a=2,\text{ b}=3\]and\[c=0\] \[\int_{0}^{\infty }{\frac{{{x}^{2}}dx}{({{x}^{2}}+4)({{x}^{2}}+9)}=\frac{\pi }{2(2+3)(3+0)(0+2)}}\] \[=\frac{\pi }{2.5.3.2}=\frac{\pi }{60}\]


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