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Math Help - definite integral #5

  1. #1
    Super Member Random Variable's Avatar
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    definite integral #5

    Show that  \int_{0}^{1} \frac{2a^{2}}{a^{2}+x^{2}} \frac{\arctan (\sqrt{2a^{2}+x^{2}})}{\sqrt{2a^{2}+x^{2}}} \ dx = \pi \arctan \Big(\frac{1}{\sqrt{2a^{2}+1}} \Big) - \Big( \arctan \frac{1}{a} \Big)^2


    If you let  a = 1 , you'll get a integral that maybe you've seen before.
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  2. #2
    Super Member Random Variable's Avatar
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    Re: definite integral #5

    I'll offer a starting point.

    Let p and q be polynomials. Then  \frac{1}{p(p+q)}} + \frac{1}{q(p+q)}} = \frac{1}{pq}

    Let  p = a^{2}+x^{2} and  q = a^{2}+t^{2}

    Then  \frac{1}{(a^{2}+x^{2})(2a^{2}+x^{2}+t^{2})} + \frac{1}{(a^{2}+t^{2})(2a^{2}+x^{2}+t^{2})} = \frac{1}{(a^{2}+x^{2})(a^{2}+t^{2})}


    Now integrate both sides over the unit square centered at  (\frac{1}{2}, \frac{1}{2})

     \int^{1}_{0} \int^{1}_{0} \frac{dx \ dt}{(a^{2}+x^{2})(2a^{2}+x^{2}+t^{2})} + \int^{1}_{0} \int^{1}_{0} \frac{dx \ dt}{(a^{2}+t^{2})(2a^{2}+x^{2}+t^{2})} = \int^{1}_{0} \int_{0}^{1} \frac{dx \ dt}{(a^{2}+x^{2})(a^{2}+t^{2})}


    If you switch the order of integration of the second integral on the left, which is permissible since the integrand is never negative, you'll notice that the the two integrals on the left are identical.


     2 \int^{1}_{0} \int^{1}_{0} \frac{dx \ dt}{(a^{2}+x^{2})(2a^{2}+ x^{2}+t^{2})} =  \int^{1}_{0} \frac{dx}{a^{2}+x^{2}} \int_{0}^{1} \frac{dt}{a^{2}+t^{2}}
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  3. #3
    Math Engineering Student
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    Re: definite integral #5

    \frac{\frac{\pi }{2}-\arctan \left( \sqrt{2{{a}^{2}}+{{x}^{2}}} \right)}{\sqrt{2{{a}^{2}}+{{x}^{2}}}}=\int_{0}^{1}  {\frac{dt}{2{{a}^{2}}+{{x}^{2}}+{{t}^{2}}}}, so we need to compute \int_{0}^{1}{\int_{0}^{1}{\frac{dx\,dt}{\left( {{a}^{2}}+{{x}^{2}} \right)\left( 2{{a}^{2}}+{{x}^{2}}+{{t}^{2}} \right)}}}.
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