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Trig and Integrals

  1. Aug 29, 2006 #1
    Trig and Integrals!!

    Hi there, this is my first post here and I'm hoping someone can help me out, I'm working on an assignment with integrals and while I can manage workng out the number versions of the question just fine I've now encountered Trig functions in integrals and I've hit a brick wall. Can anyone possibly help me get this stuff started? I need to know how to do this for my exam weeks next week. Thanks for any and all help.

    tan^5x Sec^3X dx

    Why must trigs be so painful?
  2. jcsd
  3. Aug 29, 2006 #2
    try to apply the trigonometic identity
    tan^2x +1 =sec^2x

    Also remember that the integral of secxtanx is secx and that the integral of sec^2x is tanx.

    There are actually methods which are helpful in solving trigonometic identities (at least, I was taught them when I was taking the calculus series), I can't remember them all right now. It should be in your calc book.

    hope that helps
    Last edited: Aug 29, 2006
  4. Aug 29, 2006 #3


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    Actually, my approach to almost any problem involving trig functions is to rewrite in terms of sine and cosine only!
    [tex]tan^5(x) Sec^3(X)= \frac{sin^5(x)}{cos^5(x)}\frac{1}{cos^5(x)}= \frac{sin^5(x)}{cos^8(x)}[/tex]
    Since that involves an odd power of sin(x), Rewrite the integral as
    [tex]\int\tan^5(x)sec^3(x)dx= \int\frac{sin^4(x)}{cos^8(x)}sin(x)dx=\int\frac{(1-cos^2(x))^4}{cos^8(x)}sin(x)dx[tex]
    Now, let u= cos(x) so that du= -sin(x)dx and the integral becomes
    [tex]\int\frac{(u^2- 1)^4}{u^8}du[/tex]
  5. Aug 29, 2006 #4


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    Or as island-boy has pointed out. When you need to tackle an integral with the product of odd power of tan, and a sec (the power of sec can be even or odd) function you can use the fact that:
    [tex]\frac{d}{dx} \sec x = \tan x \sec x[/tex]
    And the Trig Identity: tan2x + 1 = sec2x Or tan2x = sec2x - 1
    It goes like this:
    [tex]\tan ^ 5 x \sec ^3 x dx = \int (\tan ^ 4 x \sec ^ 2 x) (\tan x \sec x) dx[/tex]
    Now make the substitution: u = sec x, and see if you can finish the problem. :)
  6. Aug 30, 2006 #5


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    I was going to point out a stupid error, then realized I was the one who wrote it! Okay, so I'll just point out a typo!
    I wrote:
    In fact, since [itex]1- cos^2(x)= sin^2(x)[/itex], [itex]sin^4(x)= (1- cos^2(x))^2[/itex], not [itex](1- cos^2(x))^4[/itex]
    Now, let u= cos(x) so that du= -sin(x)dx and the integral becomes[tex]\int\frac{(u^2- 1)^2}{u^8}du[/tex]
  7. Sep 1, 2006 #6


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    I get

    [tex] \int \tan^{5}x \sec^{3} \ dx =-\int \frac{\left(1-u^{2}\right)^{2}}{u^{8}} \ du [/tex] with [itex] u=cos x [/itex]

    which is a little different from what HofIvy wrote.

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