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Math Help - tan sec identity

  1. #1
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    tan sec identity

    Hi so the problem is to evaluate

    integral sign ("S") tan^4 y sec^3 y dy

    now the identity says if the power of tan is odd - which it isn't - or if the power of sec is even which it isn't so what do I do with this so I can use the identity. I thought about (tan^2 y)^2 sec^2 y sec y dy but I think I am just going to end up with leftovers I don't know what to do with

    Direction is appreciated.

    Thanks
    Calculus Beginner
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  2. #2
    MHF Contributor Amer's Avatar
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    Quote Originally Posted by calcbeg View Post
    Hi so the problem is to evaluate

    integral sign ("S") tan^4 y sec^3 y dy

    now the identity says if the power of tan is odd - which it isn't - or if the power of sec is even which it isn't so what do I do with this so I can use the identity. I thought about (tan^2 y)^2 sec^2 y sec y dy but I think I am just going to end up with leftovers I don't know what to do with

    Direction is appreciated.

    Thanks
    Calculus Beginner
    \int \tan ^4 y \sec ^3 y dy

    \tan ^2 y = \sec ^2 y -1

    \int (\sec ^2y -1 )^2 \sec ^3 y dy

    \int \sec ^7 y -2\sec ^5 y + \sec ^3y\cdot dy

    in general

    \int \sec ^n y\cdot dy = \frac{1}{n-1} \sec ^{n-2} y \tan y + \frac{n-2}{n-1} \int \sec ^{n-2} y\cdot dy  , 2\leq n

    apply it more than one time if n is more than 3 in the end you will get

    \int \sec y \cdot dy and this equal

    \int \sec y \left( \frac{\sec y + \tan y }{\sec y + \tan y }\right) \cdot dy

    \int \frac{\sec ^2y + \sec y \tan y }{\sec y + \tan y } \cdot dy

    the numerator is the derivative of the denominator so the result is ln(denominator )

    \int \frac{\sec ^2y + \sec y \tan y }{\sec y + \tan y } \cdot dy= \ln (\sec y + \tan y ) + c
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