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Math Help - Please help me with a trig derivative

  1. #1
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    Please help me with a trig derivative

    I need to find the integral of this:



    Any help would be appreciated!
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  2. #2
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    Quote Originally Posted by phack View Post
    I need to find the integral of this:



    Any help would be appreciated!
    You need to find,
    \int \frac{\sec^4 \sqrt{x}}{\sqrt{x}} dx
    Let, u=\sqrt{x}
    Thus, after the substitution,
    2\int \sec^4 u du
    Now write,
    2\int \sec^2 u \sec^2 u du=2\int (\tan^2 +1)\sec^2 u du
    Let, t=\tan u then t'=\sec^2 u.
    Thus,
    2\int (t^2+1) dt=\frac{2}{3}t^3+2t+C
    Substitute,
    \frac{2}{3} \tan^3 u+2\tan u+C
    Substitute,
    \frac{2}{3} \tan^3 \sqrt{x}+2\tan \sqrt{x}+C
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