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Math Help - Integration Using Double Angle Formulas - Ex 3

  1. #1
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    Integration Using Double Angle Formulas - Ex 3

    How about this example?

    \int \tan 5x dx

    How can a double angle formula be used in this situation? hint?
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    Re: Integration Using Double Angle Formulas - Ex 3

    I would use u-sub u= 5x and write tan(u) as  \frac{sin(u)}{cos(u)}


    the double angle would be \displaystyle \int \frac{2tan(\frac{5}{2}x)}{1-tan^2(\frac{5}{2}x)}dx I'm not interested in solving it this way.
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    Re: Integration Using Double Angle Formulas - Ex 3

    One more time: u= 5x gives a very simple integral.

    (If your problem had been \int sin^2(x) dx, \int cos^4(x) dx, and \int tan^5(x) dx that would be a different matter.)
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    Re: Integration Using Double Angle Formulas - Ex 3

    How about? This is what would need double angle stuff:

    \int tan^{2}(5x) dx
    Last edited by Jason76; August 11th 2014 at 06:10 PM.
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    Re: Integration Using Double Angle Formulas - Ex 3

    This is easiest done using a Pythagorean Identity:

    $\displaystyle \begin{align*} \int{ \tan^2{(5x)}\,\mathrm{d}x} &= \int{ \sec^2{(5x)} - 1 \, \mathrm{d}x} \end{align*}$

    You should be able to go from here...
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    Re: Integration Using Double Angle Formulas - Ex 3

    K double-angle rule should be reserved for cos^2, sin^2 and cos^(2n)*sin^(2m) only.
    Tangent and secant? No, please don't use double-angle on them.
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    Re: Integration Using Double Angle Formulas - Ex 3

    Quote Originally Posted by Prove It View Post
    This is easiest done using a Pythagorean Identity:

    $\displaystyle \begin{align*} \int{ \tan^2{(5x)}\,\mathrm{d}x} &= \int{ \sec^2{(5x)} - 1 \, \mathrm{d}x} \end{align*}$

    You should be able to go from here...
    \int tan^{2}(5x) dx

    \int tan^{2}(u)dx

    \int sec^{2}(u) - 1 dx

    \dfrac{1}{5}\dfrac{\tan^{3} 5x}{3} - x + C

    \dfrac{\tan^{3} 5x}{15} - x + C

    \dfrac{\tan^{3} x}{3} - x + C
    Last edited by Jason76; August 11th 2014 at 10:32 PM.
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    Re: Integration Using Double Angle Formulas - Ex 3

    No. What is the derivative of tan(x) ?
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    Re: Integration Using Double Angle Formulas - Ex 3

    Quote Originally Posted by Jason76 View Post
    \int tan^{2}(5x) dx

    \int tan^{2}(u)dx

    \int sec^{2}(u) - 1 dx
    You really need to write these with the correct variable. The first integration is over x. Good. The remaining two are over u. So you need to use dx = (1/5) du.

    -Dan
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