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Math Help - Conceptual Integrals

  1. #1
    Junior Member Selim's Avatar
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    Conceptual Integrals

    \int h(x)f^1(x) * dx = \int

    and

    \int \frac{2}{3x^2 + 4x + 5} * dx = \int<br />
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  2. #2
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    Quote Originally Posted by Selim View Post
    \int h(x)f^1(x) * dx = \int

    and

    \int \frac{2}{3x^2 + 4x + 5} * dx = \int<br />
    your first question is unclear.

    your second question is very easy. express the denominator in the form x^2+-a^2 where a is a constant. then integrate.
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  3. #3
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    Quote Originally Posted by Selim View Post
    \int h(x)f^1(x) * dx = \int

    and

    \int \frac{2}{3x^2 + 4x + 5} * dx = \int<br />
    The bottom does not factorise, so you can't use partial fractions.

    That means you'll have to complete the square and use a trigonometric substitution.


    \int{\frac{2}{3x^2 + 4x + 5}\,dx} = \frac{2}{3}\int{\frac{1}{x^2 + \frac{4}{3}x + \frac{5}{3}}\,dx}

     = \frac{2}{3}\int{\frac{1}{x^2 + \frac{4}{3}x + \left(\frac{2}{3}\right)^2 - \left(\frac{2}{3}\right)^2 + \frac{5}{3}}\,dx}

     = \frac{2}{3}\int{\frac{1}{\left(x + \frac{2}{3}\right)^2 + \frac{11}{9}}\,dx}


    Now make the substitution x + \frac{2}{3} = \sqrt{\frac{11}{9}}\tan{\theta} = \frac{\sqrt{11}}{3}\tan{\theta}.

    Note that dx = \frac{\sqrt{11}}{3}\sec^2{\theta}\,d\theta.

    Also note that \theta = \arctan{\left[\frac{3\sqrt{11}}{11}\left(x + \frac{2}{3}\right)\right]}.


    Then the integral becomes

     = \frac{2}{3}\int{\frac{1}{\left(\frac{\sqrt{11}}{3}  \tan{\theta}\right)^2 + \frac{11}{9}}\,\frac{\sqrt{11}}{3}\sec^2{\theta}\,  d\theta}

     = \frac{2}{3}\int{\frac{\frac{\sqrt{11}}{3}\sec^2{\t  heta}}{\frac{11}{9}\tan^2{\theta} + \frac{11}{9}}\,d\theta}

     = \frac{2}{3}\int{\frac{\frac{\sqrt{11}}{3}\sec^2{\t  heta}}{\frac{11}{9}(\tan^2{\theta} + 1)}\,d\theta}

     = \frac{2\sqrt{11}}{11}\int{\frac{\sec^2{\theta}}{\s  ec^2{\theta}}\,d\theta}

     = \frac{2\sqrt{11}}{11}\int{1\,d\theta}

     = \frac{2\sqrt{11}}{11}\theta + C

     = \frac{2\sqrt{11}}{11}\arctan{\left[\frac{3\sqrt{11}}{11}\left(x + \frac{2}{3}\right)\right]} + C.
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  4. #4
    Junior Member Selim's Avatar
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    Thanks for that messy one, sir, and the other one is pretty easy,

    \int h(x) * f^1(x) * dx = h(x) * f(x) - \int f(x) * h^1(x)

    I was simply supposed to integrate it by parts.
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