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Thread: Calculus: Integration of (Tan x)^2

  1. #1

    Calculus: Integration of (Tan x)^2

    So, now I need help...

    I've been trying to figure out the integral of (Tan x)^2, but I just can't...
    The answer, from the back of the book, is Tan x - x + C.
    Can someone please show me the steps? Thanks.

  2. #2
    tan^2(x) = sec^2(x) - 1

    Sub u = tan(x)
    and du = sec^2(x)dx

    Try that, you should get the right answer, although I haven't done trigonometric integration in 2 years.

  3. #3
    wait if you change tan^2 x to sec^2 x - 1. how could you substitute u = tan x? you don't have tan left... thanks. but can anyone please help me out? thanks...

  4. #4
    You can ignore the first line I just wrote down the identity because I thought you'd need it. I'll do the integration on paper later when I get a few minutes, unless someone else gets the chance to do it first.

  5. #5
    Maxtr0sity's Avatar
    I Survived Abortion
    Simple Calc 1:
    Int of (tan x)^2 dx
    Since (secx)^2 - 1 = (tanx)^2, substitute it in so,
    Int of (secx)^2 - 1 dx
    Integrate, tanx - x + C.

  6. #6
    you mean, since d/dx of tan x = sec^2 x.
    you integrated that and got tan x ?
    so, tan^2 x is tan x - x + C?

  7. #7
    Maxtr0sity's Avatar
    I Survived Abortion
    The last line you said, the antiderivative of tangent squared x is tanx - x + C.

  8. #8
    alright, thanks. i have to go now....

  9. #9
    Bah! Max beat me to it Anyhoo, since I forgot most of my trig identities, here is another way you could solve it:
    Attached Thumbnails Attached Thumbnails Click image for larger version. 

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    Great, now even Kirupa is { facebooking | twittering }

  10. #10
    Maxtr0sity's Avatar
    I Survived Abortion
    That's pretty neat handwriting Special K.

  11. #11
    lol I was about to come back and give it a go. Is your paper actually yellow K? Did you forget Simp's advice about yellow snow?

  12. #12
    oh my god kirupa. you're such a nice person man... thanks everybody...

    i actually got [intg]sec^2 - 1 dx
    but, i didn't know where to go from there.

    again, thank you very much everybody.

    btw, kirupa tan^2 x = sec^2 x - 1. hehe. (i'm sure you already figured that out )

  13. #13
    aye, apparently it seems my brain was still working at 3am in the morning... you won't know what I'm talking about, but if the mods can see deleted posts they'll notice that I posted to try tanx = sinx/cosx

    right.. enough of my gloating.

  14. #14
    Wow that's some neat writing Kirupa. Not neat handwriting, but very nice organization - numbering lines, semi-colons (just like coding!), square brackets, etc.
    If you notice this notice you will notice that this notice is not worth noticing.

    "Are you doing anything tonight? If not, how about me?"

    Opera Sucks! - FIX IT
    Oliver Zheng

  15. #15
    I added the line numbering because it was going to be posted online. I single-space and write much smaller than that

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