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Thread: Integral Multiplication

  1. #1
    Senior Member polymerase's Avatar
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    Integral Multiplication

    $\displaystyle \int tan\;x\sqrt{1+sec^4x}\;dx$

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  2. #2
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    I think you first need to get rid of that square root by doing some substitution.

    Try $\displaystyle tan \theta = \sec^2 (x)$. It might not actually work out...but that 4th power has to be removed before you can replace that $\displaystyle \sqrt{1 + (\tan \theta)^2}$ with just $\displaystyle \sec \theta$.

    At least that's what I would try. You just shouldn't use integration by parts if it looks like you can still substitute.
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  3. #3
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by TrevorP View Post
    I think you first need to get rid of that square root by doing some substitution.

    Try $\displaystyle tan \theta = \sec^2 (x)$. It might not actually work out...but that 4th power has to be removed before you can replace that $\displaystyle \sqrt{1 + (\tan \theta)^2}$ with just $\displaystyle \sec \theta$.

    At least that's what I would try. You just shouldn't use integration by parts if it looks like you can still substitute.
    not sure that would make life any easier, after all, what would $\displaystyle \tan x$ become? $\displaystyle \sqrt{\tan \theta - 1}$. worst, what would $\displaystyle d \theta$ be? i never wrote anything down, but just thinking about it seems like a mess to me. unless you have some trick to simplify things
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  4. #4
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    Quote Originally Posted by polymerase View Post
    $\displaystyle \int tan\;x\sqrt{1+sec^4x}\;dx$
    Okay, this will be a mess. It takes lots of substitutions, but having patience makes sense.

    First rewrite your integral

    $\displaystyle \int {\sqrt {1 + \sec ^4 x} \tan x\,dx} = \int {\sqrt {1 + \cos ^4 x} \sec ^2 x\tan x\,dx} .$

    Substitute $\displaystyle u=\tan x,$ the integral becomes

    $\displaystyle \int {\frac{{u\sqrt {1 + \left( {u^2 + 1} \right)^2 } }}
    {{u^2 + 1}}\,du} .$

    Make another substitution according to $\displaystyle v=u^2+1,$ the integral becomes

    $\displaystyle \frac{1}
    {2}\int {\frac{{\sqrt {1 + v^2 } }}
    {v}\,dv} .$

    Now a final substitution defined by $\displaystyle \varphi ^2 = 1 + v^2.$ (We're gettin' close.)

    $\displaystyle \frac{1}
    {2}\int {\frac{{\varphi ^2 }}
    {{\varphi ^2 - 1}}\,d\varphi } = \frac{1}
    {2}\left[ {\int {d\varphi } + \frac{1}
    {2}\left( {\int {\frac{1}
    {{\varphi - 1}}\,d\varphi } - \int {\frac{1}
    {{\varphi + 1}}\,d\varphi } } \right)} \right].$

    Now this is easy, so we have

    $\displaystyle \frac{1}
    {2}\int {\frac{{\varphi ^2 }}
    {{\varphi ^2 - 1}}\,d\varphi } = \frac{1}
    {2}\left( {\varphi + \frac{1}
    {2}\ln \left| {\frac{{\varphi - 1}}
    {{\varphi + 1}}} \right|} \right) + k.$

    After back substitute we finally-happily get that

    $\displaystyle \int {\sqrt {1 + \sec ^4 x} \tan x\,dx} = \frac{1}
    {2}\left( {\frac{{\sqrt {1 + \cos ^4 x} }}
    {{\cos ^2 x}} + \frac{1}
    {2}\ln \left| {\frac{{\sqrt {1 + \cos ^4 x} - \cos ^2 x}}
    {{\sqrt {1 + \cos ^4 x} + \cos ^2 x}}} \right|} \right) \,+ \,k.$
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  5. #5
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    $\displaystyle \tan \theta = \sec^2 x$
    $\displaystyle \sec^2 \theta d \theta = 2 \sec x \sec x \tan x dx$
    $\displaystyle \sec^2 \theta d \theta = 2 \tan \theta \tan x dx$
    $\displaystyle \frac{\sec^2 \theta d \theta}{2 \tan \theta \tan x} = dx$

    $\displaystyle \int \frac{\tan x \sqrt{1 + \tan^2 \theta} \sec^2 \theta d \theta }{2 \tan \theta \tan x} = \tfrac{1}{2} \int \frac{\sec^3 \theta d \theta}{\tan \theta }$

    Now use integration by parts to solve that.

    EDIT: Ok Krizalid already gave you his answer I guess I'll finish this.

    $\displaystyle = \tfrac{1}{2} \int \underbrace{\sec^2 \theta}_{\tfrac{d}{d \theta} \tan \theta} \underbrace{\frac{\sec \theta}{\tan \theta }}_{\csc \theta} d \theta = \tfrac{1}{2} \Big( \tan \theta \csc \theta - \int \tan \theta (-) \csc \theta \cot \theta d \theta \Big)$
    $\displaystyle = \tfrac{1}{2} \Big( \sec \theta + \int \csc \theta d \theta \Big) = \tfrac{1}{2} \Big( \sec \theta - ln|\csc \theta + \cot \theta| \Big)$
    $\displaystyle = \tfrac{1}{2} \Big( \sec ( \arctan ( \sec^2 x ) ) - ln|\csc ( \arctan ( \sec^2 x ) ) + \cot ( \arctan ( \sec^2 x ) )| \Big)$
    Last edited by TrevorP; Jan 30th 2008 at 05:03 PM.
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  6. #6
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Krizalid View Post
    Okay, this will be a mess. It takes lots of substitutions, but having patience makes sense.

    First rewrite your integral

    $\displaystyle \int {\sqrt {1 + \sec ^4 x} \tan x\,dx} = \int {\sqrt {1 + \cos ^4 x} \sec ^2 x\tan x\,dx} .$

    Substitute $\displaystyle u=\tan x,$ the integral becomes

    $\displaystyle \int {\frac{{u\sqrt {1 + \left( {u^2 + 1} \right)^2 } }}
    {{u^2 + 1}}\,du} .$

    Make another substitution according to $\displaystyle v=u^2+1,$ the integral becomes

    $\displaystyle \frac{1}
    {2}\int {\frac{{\sqrt {1 + v^2 } }}
    {v}\,dv} .$

    Now a final substitution defined by $\displaystyle \varphi ^2 = 1 + v^2.$ (We're gettin' close.)

    $\displaystyle \frac{1}
    {2}\int {\frac{{\varphi ^2 }}
    {{\varphi ^2 - 1}}\,d\varphi } = \frac{1}
    {2}\left[ {\int {d\varphi } + \frac{1}
    {2}\left( {\int {\frac{1}
    {{\varphi - 1}}\,d\varphi } - \int {\frac{1}
    {{\varphi + 1}}\,d\varphi } } \right)} \right].$

    Now this is easy, so we have

    $\displaystyle \frac{1}
    {2}\int {\frac{{\varphi ^2 }}
    {{\varphi ^2 - 1}}\,d\varphi } = \frac{1}
    {2}\left( {\varphi + \frac{1}
    {2}\ln \left| {\frac{{\varphi - 1}}
    {{\varphi + 1}}} \right|} \right) + k.$

    After back substitute we finally-happily get that

    $\displaystyle \int {\sqrt {1 + \sec ^4 x} \tan x\,dx} = \frac{1}
    {2}\left( {\frac{{\sqrt {1 + \cos ^4 x} }}
    {{\cos ^2 x}} + \frac{1}
    {2}\ln \left| {\frac{{\sqrt {1 + \cos ^4 x} - \cos ^2 x}}
    {{\sqrt {1 + \cos ^4 x} + \cos ^2 x}}} \right|} \right) + k.$
    nice.

    could you have dealt with something like $\displaystyle \int \frac {\sqrt{1 + u^4}}u~du$ in the same way? because we could have gotten that by one substitution, $\displaystyle u = \sec x$
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  7. #7
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    Quote Originally Posted by Jhevon View Post
    could you have dealt with something like $\displaystyle \int \frac {\sqrt{1 + u^4}}u~du$ in the same way? because we could have gotten that by one substitution, $\displaystyle u = \sec x$
    Sure, just substitute $\displaystyle v^2=1+u^4\implies v\,dv=2u^3\,du,$ then

    $\displaystyle \int {\frac{{\sqrt {1 + u^4 } }}
    {u}\,du} = \frac{1}
    {2}\int {\frac{{v^2 }}
    {{v^2 - 1}}\,dv} ,$ and the rest follows.

    (Of course you meant to say $\displaystyle u^2=\tan x.$ )
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