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Math Help - Check the answer of integration(Definite)

  1. #1
    Like a stone-audioslave ADARSH's Avatar
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    Smile Check the answer of integration(Definite)

    The value of \int_0^{\infty}\{\frac{dx}{(a^2+x^2)^7}\} is equal to ?

    My try:


    Put  x=a tan(\theta)

    dx =asec^2(\theta)d\theta

    Thus integration becomes

    \int_{0}^{\frac{\pi}{2}}\frac{a sec^2(\theta)d\theta}{a^{14}~(1+tan^2(\theta))^7}

    Using 1+tan^2(h) = sec^2(h)


    \frac{1}{a^{13}}\int_{0}^{\frac{\pi}{2}}\{\frac{d\  theta}{(sec^{12}(\theta)}\}

    \frac{1}{a^{13}}\int_{0}^{\frac{\pi}{2}}\{cos^{12}  (\theta)d\theta\}

    Now here I have used the formula that for even "n"

    \int_{0}^{\pi /2} (cos^{n}(xdx)) =  \frac{n-1}{n} \cdot \frac{n-3}{n-2} ....\frac{1}{2}\cdot \frac{\pi}{2}

    \frac{1}{a^{13}}(\frac{11}{12}\cdot \frac{9}{10}...\frac{1}{2}\cdot \frac{\pi}{2})

    My answer thus was \frac{363}{2048a^{13}}


    Options are


    • \frac{231}{2048a^{13}}


    • \frac{235}{2048a^{13}}


    --------------------------
    I don't have the answer.
    Where did I go wrong.
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  2. #2
    MHF Contributor Calculus26's Avatar
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    solution

    The correst answer is *pi verified by Mathcad and by hand

    method is good -- 11/12*9/10*7/8*5/6*3/4*1/2*π/2 = *pi
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  3. #3
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by ADARSH View Post
    The value of \int_0^{\infty}\{\frac{dx}{(a^2+x^2)^7}\} is equal to ?

    My try:


    Put  x=a tan(\theta)

    dx =asec^2(\theta)d\theta

    Thus integration becomes

    \int_{0}^{\frac{\pi}{2}}\frac{a sec^2(\theta)d\theta}{a^{14}~(1+tan^2(\theta))^7}

    Using 1+tan^2(h) = sec^2(h)


    \frac{1}{a^{13}}\int_{0}^{\frac{\pi}{2}}\{\frac{d\  theta}{(sec^{12}(\theta)}\}

    \frac{1}{a^{13}}\int_{0}^{\frac{\pi}{2}}\{cos^{12}  (\theta)d\theta\}

    Now here I have used the formula that for even "n"

    \int_{0}^{\pi /2} (cos^{n}(xdx)) =  \frac{n-1}{n} \cdot \frac{n-3}{n-2} ....\frac{1}{2}\cdot \frac{\pi}{2}

    \frac{1}{a^{13}}(\frac{11}{12}\cdot \frac{9}{10}...\frac{1}{2}\cdot \frac{\pi}{2})

    My answer thus was \frac{363}{2048a^{13}}{\color{red}\,\pi}


    Options are


    • \frac{231}{2048a^{13}}


    • \frac{235}{2048a^{13}}


    --------------------------
    I don't have the answer.
    Where did I go wrong.
    First, You missed a \pi in your solution.

    Your mistake is in the line before you told us your answer. When you multiply it out, you should have \frac{1}{a^{13}}\frac{11\cdot9\cdot7\cdot5\cdot3\c  dot1\cdot\pi}{12\cdot10\cdot8\cdot6\cdot4\cdot2\cd  ot2}=\boxed{\frac{231\pi}{a^{13}2048}}
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  4. #4
    Like a stone-audioslave ADARSH's Avatar
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    Quote Originally Posted by Calculus26 View Post
    The correst answer is *pi verified by Mathcad and by hand

    method is good -- 11/12*9/10*7/8*5/6*3/4*1/2*π/2 = *pi
    My calculated answer was wrong its supposed to be

    \frac{363}{1024a^{13}}


    Taking pi as 22/7, The answer will be correct now . thanks

    Chris:No I didn't miss the pi, it was to be converted in accordance to the options. Though I messed up the calculation part.


    So both the options are incorrect as "pi" is missing in them. Thanks to both of you for verifying

    Adarsh
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