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Thread: Integral of Sec^3 x tan x dx by sub

  1. June 4th 2008, 12:25 AM #1
    Craka
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    Integral of Sec^3 x tan x dx by sub

    Hey guys, have another one I hope you can help me with trying to solve this by substitution. If I let u = tan x , than du = sec^2 x dx . Would it be correct that then dx = du / sec^2 x And now at this point I'm lost
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  2. June 4th 2008, 12:34 AM #2
    Chris L T521
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    Quote Originally Posted by Craka View Post
    Hey guys, have another one I hope you can help me with trying to solve this by substitution. If I let u = tan x , than du = sec^2 x dx . Would it be correct that then dx = du / sec^2 x And now at this point I'm lost
    There is a little trick to this one:

    \int \sec^3(x)\tan(x)\,dx=\int \sec^2(x)\cdot(\sec(x)\tan(x))\,dx.

    Let u=\sec(x) \implies \,du=\sec(x)\tan(x)\,dx


    Now the integral becomes: \int u^2\,du=\frac{1}{3}u^3+C=\color{red}\boxed{\frac{1 }{3}\sec^3(x)+C}

    Hope this makes sense!
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  3. June 4th 2008, 12:53 AM #3
    Chris L T521
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    Here's something that will help you out in the future:

    If we have an integral in the form of \int \sin^m(x)\cos^n(x)\,dx, and

    n is odd :
    • split off a factor of \cos(x)
    • apply the identity \cos^2(x)=1-\sin^2(x)
    • apply the substitution u=\sin(x)
    m is odd :
    • split off a factor of \sin(x)
    • apply the identity \sin^2(x)=1-\cos^2(x)
    • apply the substitution u=\cos(x)
    m is even and n is even :
    • use \sin^2(x)=\frac{1}{2}(1-\cos(2x)) and \cos^2(x)=\frac{1}{2}(1+\cos(2x)) to lower the powers of cosine and sine.




    If we have an integral in the form of \int \tan^m(x)\sec^n(x)\,dx, and

    n is even :
    • split off a factor of \sec^2(x)
    • apply the identity \sec^2(x)=\tan^2(x)+1
    • apply the substitution u=\tan(x)
    m is odd :
    • split off a factor of \sec(x)\tan(x)
    • apply the identity \tan^2(x)=\sec^2(x)-1
    • apply the substitution u=\sec(x)
    m is even and n is odd :
    • use \tan^2(x)=\sec^2(x)-1 to get the equation ins terms of secant. Then apply reduction formulas.




    If we have an integral in the form of \int \cot^m(x)\csc^n(x)\,dx, and

    n is even :
    • split off a factor of \csc^2(x)
    • apply the identity \csc^2(x)=\cot^2(x)+1
    • apply the substitution u=\cot(x)
    m is odd :
    • split off a factor of \csc(x)\cot(x)
    • apply the identity \cot^2(x)=\csc^2(x)-1
    • apply the substitution u=\csc(x)
    m is even and n is odd :
    • use \cot^2(x)=\csc^2(x)-1 to get the equation ins terms of cosecant. Then apply reduction formulas.


    Hope this will help you out in future problems!!!
    Last edited by Chris L T521; October 3rd 2008 at 08:44 PM.
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  4. June 4th 2008, 01:00 AM #4
    Craka
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    Thanks for that.
    Seems to be alot of little tricks with integration.
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