CHAPTER N 
REDUCTION FORMULAS 
1. 
The Integral 
sin" x cos m x dx. 
We begin by taking the differential of a function of the same 
type as the integrand : 
(1) d (sin* x cos'* x) = v sin* -1 x cos'** 1 xdx — /J. sin* +1 x cos 1 ^ 1 x dx. 
If we integrate each side of this equation, we obtain a relation 
between the integrals 
Thus if we wished, in the given integral, to increase n and decrease 
m (or vice versa), we could effect the result. But this is a very 
special and relatively unimportant case. It does not enable us to 
change one of the exponents without changing the other. We are 
led, therefore, to make a trigonometric reduction. Write 
cos'* +1 x = cos'* -1 x cos 2 x = cos»* -1 x — cos»* -1 x sin 2 x. 
Then equation (1) becomes : 
(2) d (sin* x cos'* x) = v sin* -1 x cos'* -1 x dx — (^ + v ) sin* +1 x cos'* -1 x dx. 
On integrating this equation, we obtain a formula whereby the 
exponent of the cosine factor is unchanged, but the exponent of the 
sine factor is changed by 2. Let 
V + 1 = 71, 
/ii — l = m. 
Then 
sin” -1 X COS m+1 X 
(3) 
n + m n + m, 
33 i 
sin” -2 x cos”* x dx.