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Homework Help: Calculus II - Trigonometric Integrals

  1. Aug 1, 2011 #1
    1. The problem statement, all variables and given/known data

    Evaluate
    integral csc(x)^4/cot(x)^2 dx

    2. Relevant equations

    3. The attempt at a solution

    Apparently I'm doing something wrong, what I'm not sure, thanks for any help

    My Answer: 2*tan(x) - (sec(x)^2*tan(x))/3 + c

    integral csc(x)^4/cot(x)^2 dx

    used fact that csc(x) = 1/sin(x)
    used fact that cot(x) = 1/tan(x)
    used fact that tan(x) = sin(x)/cos(x)

    integral sin(x)^2/(sin(x)^4*cos(x)^2) dx

    used the fact that x^m/x^n = x^(m-n)
    used the fact that x^-m = 1/x^m

    integral dx/(sin(x)^2*cos(x)^2)

    used the fact that sin(x)^2 + cos(x)^2 = 1
    solved for sin(x)^2
    sin(x)^2 = 1 - cos(x)^2

    integral dx/((1-cos(x)^2)*cos(x)^2)

    expanded

    integral dx/(cos(x)^2 - cos(x)^4)

    broke into two separate integrals

    integral dx/cos(x)^2 - integral dx/cos(x)^4

    used fact that 1/cos(x) = sec(x)

    integral sec(x)^2 dx - integral sec(x)^4 dx

    begun evaluating first integral using
    integral sec(x)^n = ( sec(x)^(n-2)*tan(x) )/( n-1 ) + ( n-2)/(n-1)*integral sec(x)^(n-2) dx, n =/= 1

    (sec(x)^(2-2)*tan(x))/(2-1) - (2-2)/(2-1) integral sec(x)^(2-2) dx - integral sec(x)^4 dx

    simplified

    tan(x) - integral sec(x)^4 dx

    begun evaluating second integral

    tan(x) - [(sec(x)^(4-2)*tan(x))/(4-1) - integral sec(x)^2 dx]

    simplified

    tan(x) - (sec(x)^2*tan(x))/3 + integral sec(x)^2 dx

    established already that integral sec(x)^2 dx = tan(x)

    2*tan(x) - (sec(x)^2*tan(x))/3 + c

    Wolfram Alpha Answer: -2 cot(2 x)+constant

    MATLAB Answer: tan(x) - 1/tan(x)

    Back of the Book Answer: tan(x) - cot(x) + c

    My answer is not equivalent
    2*tan(5) - (sec(5)^2*tan(5))/3 is about 7.243183523
    tan(5) - 1/tan(5) is about -3.084702091
    -2 cot(2*5) is about -3.084702091
    tan(5) - cot(5) is about -3.084702091
     
  2. jcsd
  3. Aug 1, 2011 #2
    The way you broke up the fraction is incorrect, you can't expand a minus sign in the denominator as you did. I suggest you start over, and this time instead expand the numerator with the identity csc2(x)=1+cot2(x). You will then find this problem to be significantly easier.
     
  4. Aug 1, 2011 #3

    SammyS

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    At this point use the double angle formula for sine: sin(2x) = 2sin(x)*cos(x).
     
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