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Thread: 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,
    $\displaystyle \int \frac{\sec^4 \sqrt{x}}{\sqrt{x}} dx $
    Let, $\displaystyle u=\sqrt{x}$
    Thus, after the substitution,
    $\displaystyle 2\int \sec^4 u du$
    Now write,
    $\displaystyle 2\int \sec^2 u \sec^2 u du=2\int (\tan^2 +1)\sec^2 u du$
    Let, $\displaystyle t=\tan u$ then $\displaystyle t'=\sec^2 u$.
    Thus,
    $\displaystyle 2\int (t^2+1) dt=\frac{2}{3}t^3+2t+C$
    Substitute,
    $\displaystyle \frac{2}{3} \tan^3 u+2\tan u+C$
    Substitute,
    $\displaystyle \frac{2}{3} \tan^3 \sqrt{x}+2\tan \sqrt{x}+C$
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