Associated Topics || Dr. Math Home || Search Dr. Math

### 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/
```
Associated Topics:
College Calculus

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search