calculus

Find the indefinite integral of 1-tanx/1+tanx

Dont know how to really approach this question. Should i use identities, or is there a power series i can use?

  1. 👍 0
  2. 👎 0
  3. 👁 235
  1. There are standard formulae for integrals of rational functions of trigonometric formulae. In this case, you can simplify things as follows.

    Let's use the abbreviation:

    t = tan(x)

    s = sin(x)

    c = cos(x)

    We can write:

    (1-t)/(1+t) =

    (c-s)/(c+s) =

    (c - s)^2/(c^2 - s^2) =

    (c^2 + s^2 - 2cs)/(c^2 - s^2)

    Then use the trigonometric identities:

    c^2 + s^2 = 1

    2 cs = sin(2x)

    c^2 - s^2 = cos(2x)

    to obtain:

    (1-t)/(1+t) =

    1/(cos(2x)) - tan(2x)

    Integrating tan(2x) is trivial. You can integrate 1/cos(2x) e.g. by putting
    x = (pi/4 - u), so that cos(2x) =
    sin(2u. Then

    1/sin(2u) =

    [cos^2(u) + sin^2(u)]/[2 sin(u)cos(u)] =

    1/2 [cot(u) + tan(u)]

    which is trivial to integrate.

    1. 👍 0
    2. 👎 0
  2. how did you get from (c-s)/(c+s) to

    (c - s)^2/(c^2 - s^2)?

    1. 👍 0
    2. 👎 0
  3. Multiply numerator and denominator by (c-s).

    1. 👍 0
    2. 👎 0
  4. thanks

    1. 👍 0
    2. 👎 0

Respond to this Question

First Name

Your Response

Similar Questions

  1. Trigonometry

    Given that a^2+b^2=2 and that (a/b)= tan(45degee+x), find a and b in terms of sinx and cosx. I don't know what i'm supposed to do, and i don't come to an answer! Help, thanks! my workings: tan(45+x)= (1+tanx)/(1-tanx) a/b =

    asked by Marrion on September 27, 2009
  2. Trigo

    Given that a^2+b^2=2 and that (a/b)= tan(45degee+x), find a and b in terms of sinx and cosx. I don't know what i'm supposed to do, and i don't come to an answer! Help, thanks! my workings: tan(45+x)= (1+tanx)/(1-tanx) a/b =

    asked by Marrion on September 26, 2009
  3. Calc

    Evaluate the integral using any method: (Integral)sec^3x/tanx dx I started it out and got secx(1tan^2x)/tanx. I know I just have to continue simplifying and finding the integral, but I'm stuck on the next couple of steps. Also, I

    asked by H on October 1, 2012
  4. Math

    1)A piano tuner uses a tuning fork. If middle C has a frequency of 264 vibrations per second, write an equation in the form d=sinw(t) for the simple harmonic motion. 2) Verify the identity tan^2X-cot^2X/tanX+cotX=tanX-cotX I'm not

    asked by Lisa on October 9, 2006
  5. Calc

    Hello im trying to integrate tan^3 dx i have solved out the whole thing but it doesnt match up with the solution.. this is what i did: first i broke it up into: integral tan^2x (tanx) dx integral (sec^2x-1)(tanx) dx then i did a u

    asked by UCI Student on May 30, 2012
  6. Calc

    Hello im trying to integrate tan^3 dx i have solved out the whole thing but it doesnt match up with the solution.. this is what i did: first i broke it up into: integral tan^2x (tanx) dx integral (sec^2x-1)(tanx) dx then i did a u

    asked by ***UCI Student*** on May 30, 2012
  7. Math

    I can't find the integral for (tanx)^(6)*(secx)^(2) I tried splitting up tanx into (tanx)^2*(tanx)^4 and let the latter equal (secx)^2 - 1. Please help, thanks!

    asked by Anonymous on February 15, 2013
  8. Math

    Evaluate the integral of (secx)^2 * (tanx)^3 *dx I started out with letting u=secx and du=secx*tanx*dx , but then I am kind of stuck because I don't know how to factor out just secx*tanx? The answer in the book said that it is

    asked by Dave on September 13, 2017
  9. Calculus

    Evaluate the indefinite integral. ∫ sec^3x tanx dx I let u=tanx took derivative.. du=sec^2x dx ..now what??

    asked by Anonymous on February 16, 2011
  10. Trigonometry

    For this question, they want me to use fundamental trig identities to simplify the expression. The problem is as follows; (tanx/csc^2x + tanx/sec^2x)(1+tanx/1+cotx) - 1/cos^2x I got as far as this; tanx(1/csc^2x +

    asked by CodyJinks on July 18, 2019

More Similar Questions