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How can I solve this without using Reduction Formula?

  1. Jul 12, 2016 #1
    1. The problem statement, all variables and given/known data
    integral of cot^2 x / csc^8 x dx

    2. Relevant equations
    u = cot x
    du = csc^2 x du

    3. The attempt at a solution
    if I use reduction formula I could answer this but it's going to be very very LONG SOLUTION

    I just need some basic integral work.
     
  2. jcsd
  3. Jul 12, 2016 #2
    First convert the expression to only sin and cos and then think of proper substitution. If youdo not wish to use reduction formula you need to juggle up the expressions and remove powers to get expression in terms of multiple angles.
     
  4. Jul 12, 2016 #3

    Ray Vickson

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    Science Advisor
    Homework Helper

    Write the integrand as ##f(x) = \cos^2(x) \sin^6 (x)##. Use ##\cos^2(x) = 1 -\sin^2(x)## to get your integral ##F = \int f(x) \, dx## in the form ##F = I_6-I_8##, where ##I_n = \int \sin^n(x) \, dx##.

    Apply integration by parts to ##I_n##, using ##u = \sin^{n-1}(x)## and ##dv = \sin(x) \, dx##. This gives
    [tex] I_n = -\cos(x) \sin^{n-1}(x) + (n-1) \int \cos^2(x) \sin^{n-2}(x) \, dx = -\cos(x) \sin^{n-1}(x) + (n-1) [I_{n-2} - I_n] [/tex]
    This is an equation connecting ##I_n## to ##I_{n-2}##, so you can solve it to express ##I_n## in terms of ##\sin(x), \cos(x)## and ##I_{n-2}##. Finally, you can express ##I_8## in terms of ##I_6##, then ##I_6## in terms of ##I_4##, etc. The answer you want will drop out pretty quickly and easily.
     
  5. Jul 25, 2016 #4
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