e case that 
ire always 
vision has 
not zero- 
Thus no 
3t lead to 
meaning, 
s m ~ 2 xdx. 
ith 
rnding 
x and 
xdx. 
2. The Integral 
REDUCTION FORMULAS 
dx 
/ 
(a 2 + x 2 ) n 
3. The Integral 
f 
It would be a false analogy with the example of the preceding 
paragraph to start with d(a 2 + x 2 )~ n , since the result would be but a 
single term, and that not yielding an integral of useful type. We 
need a product, and Exercises 6 and 7 in § 1 suggest the plan: * 
(1) d [x(a 2 + x 2 )~ m ~\ = (a 2 + £ 2 )~ m dx —2 mx 2 (a 2 -f- x 2 )~ m ~' i dx. 
In the last term, write the factor x 2 in the form: 
x 2 = (a 2 + x 2 ) — a 2 . 
Thus 
(2) d |>(a 2 + a 2 )- m ] = (1 - 2 m) (a 2 + x 2 Y m dx - 2 m a 2 (a 2 + x 2 )- m ~ l dx. 
It is now clear that, on setting m + 1 = n and integrating, we 
shall have a reduction formula worth while, namely: 
/ox C dx x ■ 2w — 3 r 
K } J (a 2 + x 2 ) n 2(m - l)a 2 (a 2 + x 2 ) n ~' (2n - 2)a 2 J 
It is this formula which occupies a pivotal position in the proof 
that every rational function can be integrated. 
EXERCISE 
In Formula (3) set x = a tan 6. Hence show how (3) can be de 
duced from the formula of Exercise 2, § 1. 
V« + bx -+- cx 2 
y = Va + bx -f- 
Let 
and take d (x m y) : 
( \_7) _1_ C'V\ , V rn 
(1) d (x m y) = mx m ~ l y dx -f- ^^— dx, 
mx m-x v = nM m -\a + bx + cx 2 ) 
y 
* There is no inductive treatment possible in this part of integration. Imagi 
nation, resourcefulness, and the power of keen observation are the qualities 
required.