A Reduction Formula and a Special Method

Date: 1/23/96 at 10:47:40
From: Anonymous
Subject: MATH

Hello.  I'm Burcak and I'm sending you two questions:

1 ) F(X)=SEC(4X)^5  I'm asking for the integration of F(X).
    ('^5' is used for the fifth power of F(X) )

2)  F(X)=(2SIN(X)+3COS(X))/(3SIN(X)+2COS(X))
    Again, I'm asking for the integration of F(X).


                                          Thanks!

Date: 7/22/96 at 22:0:42
From: Doctor Jerry
Subject: Re: MATH

A convenient formula (called a reduction formula) for solving 
questions like (1) is:

int(sec^n x dx)=(1/(n-1)) sec^(n-2) x tan(x) + 
                 ((n-2)/(n-1)) int( sec^(n-2) x dx.

n is a positive number, greater than or equal to 3.  The idea is to 
use the formula repeatedly, until the n-2 becomes either 1 or 2.  The 
integral of secant to the first or second powers is a standard, 
known formula.  First, however, make the substitution 4x=w in the 
original integral.

Question (2) is more interesting.  Although it is a rational function 
of sin x and cos x and can be solved by the well known substitution 
u=tan(x/2), it can also be solved by a special method.

First, note that

(2sin x+3cos x)/(3sin x+2cos x)+K = 
                 ((2+3K)sin x+(3+2K)cos x)/(3sin x+2cos x).

So that the numerator is a numerical multiple of the derivative of 
the denominator, we try to find L so that
2+3K=-2L and 3+2K=3L.

The solution of these equations is K = -12/13 and L = 5/13.

From this it follows that the integral of your F(X) is
(12X)/13+ 5LN(2COS X+3SIN x)/13.


-Doctor Jerry,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/