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Math Help - integration

  1. #1
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    integration

    hi, is there any method of integrating a hyperbolic or trignometric function whos degree is 4, such as (tanx)^4, (cosechx)^4 WITHOUT resorting to reduction formulae??
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  2. #2
    TD!
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    Sure, there just isn't a "golden recipe" which will be succesfull every time. You'll need to play arround with trig formulas. I'll do one as an example.

    <br />
\int {\tan ^4 x} dx = \int {\tan ^2 x\tan ^2 x} dx = \int {\tan ^2 x\left( {\sec ^2 x - 1} \right)} dx<br />

    Now I'll split the integral in two. The first one is easy since the derivative of tan(x) is exactly secē(x), so:

    <br />
\int {\tan ^2 x\sec ^2 x} dx = \int {\tan ^2 x} d\left( {\tan x} \right) = \frac{{\tan ^3 x}}{3} + C<br />

    For the second one; I convert to sin(x) and cos(x):

    <br />
\int { - \tan ^2 x} dx =  - \int {\frac{{\sin ^2 x}}{{\cos ^2 x}}} dx = \int {\frac{{\cos ^2 x - 1}}{{\cos ^2 x}}} dx<br />

    Splitting the integral in two again gives us simply:

    <br />
\int {dx}  - \int {\frac{1}{{\cos ^2 x}}} dx = x - \tan x + C<br />

    So we conclude, without reduction-formula:

    <br />
\int {\tan ^4 x} dx = \frac{{\tan ^3 x}}{3} - \tan x + x + C<br />
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  3. #3
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    reply

    thanks, that was great help..i noticed the reduction formula gave effectively the same answer but in a different form.
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  4. #4
    TD!
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    Yes, that's perfectly possible and happens very often when you compare 'manual integration' and integration through such reduction formulas.
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