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Help with integral from Apostol Calculus

  1. Apr 28, 2014 #1
    Help with integral from Apostol "Calculus"

    1. The problem statement, all variables and given/known data
    I seem to be stuck trying to prove the following integral from Apostol "Calculus" Volume 1 Section 5.11, Question 33.
    [itex]
    \int\frac{\cos^mx}{\sin^nx}dx = -\frac{\cos^{m+1}x}{(n-1)\sin^{n-1}x}-\frac{m-n+2}{n-1}\int\frac{\cos^mx}{\sin^{n-2}x} dx + C\,\,(n \neq 1)
    [/itex]


    2. Relevant equations
    N/A


    3. The attempt at a solution
    My thinking so far has been that if I take
    [itex]
    I = \int\frac{\cos^mx}{\sin^nx}dx
    [/itex]
    I have been able to prove that
    [itex]
    I = -\frac{\cos^{m-1}x}{(n-1)\sin^{n-1}x} - \frac{m-1}{n-1}\int\frac{\cos^{m-2}x}{\sin^{n-2}x}\,dx+C\,\,\,\,\,(1)
    [/itex]
    and
    [itex]
    I = \frac{\cos^{m-1}x}{(m-n)\sin^{n-1}x} + \frac{m-1}{m-n}\int\frac{\cos^{m-2}x}{\sin^nx}\,dx+C\,\,\,\,\,(2)
    [/itex]
    but showing that
    [itex]
    I = -\frac{\cos^{m+1}x}{(n-1)\sin^{n-1}x}-\frac{m-n+2}{n-1}\int\frac{\cos^mx}{\sin^{n-2}x} dx + C
    [/itex]
    seems to be eluding me. I attempted to apply a similar technique what I used on [itex](1)[/itex] to get [itex](2)[/itex] to try to obtain this integral, but it didn't seem to work.
    I can also show that
    [itex]
    I = -\frac{\cos^{m+1}x}{(m+1)\sin^{n+1}x} - \frac{n+1}{m+1}\int\frac{\cos^{m+2}x}{\sin^{n+2}x}\, dx + C
    [/itex]
    but there's obviously more to it from this perspective.
    Any help would be greatly appreciated.
     
  2. jcsd
  3. Apr 28, 2014 #2

    Zondrina

    User Avatar
    Homework Helper

    Perhaps you could write the integral as:

    ##\int cos^m(x)sin^{-n}(x) dx## and apply an existing reduction formula.
     
  4. May 1, 2014 #3

    lurflurf

    User Avatar
    Homework Helper

    You need to use $1=\sin^2(x)+\cos^2(x)$

    $$\int \! \frac{\cos^{m}(x)}{\sin^{n}(x)} \, \mathrm{d}x=
    \int \! \frac{\cos^{m}(x)}{\sin^{n}(x)}(\sin^2(x)+\cos^2(x)) \, \mathrm{d}x=
    \\
    \int \! \frac{\cos^{m}(x)}{\sin^{n}(x)}\sin^2(x) \, \mathrm{d}x+
    \int \! \frac{\cos^{m}(x)}{\sin^{n}(x)}\cos^2(x) \, \mathrm{d}x=
    \\
    \int \! \frac{\cos^{m}(x)}{\sin^{n-2}(x)} \, \mathrm{d}x+
    \int \! \frac{\cos^{m+2}(x)}{\sin^n(x)} \, \mathrm{d}x$$

    Integrate the bit with m+2 by parts to reach the desired form.
     
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