40 CALCULUS 
evaluated, and so the formulas are important chiefly in the case that 
n and m are integers. 
Formula (2) and the corresponding integral relation are always 
true. In passing, however, to the later formulas, a division has 
taken place, and it is tacitly assumed that the divisor is not zero- 
If it were, the resulting formula would have no meaning. Thus no 
danger can arise, for a formula that has no meaning cannot lead to 
a wrong result. Whenever one of these formulas has a meaning, 
it is correct. 
EXERCISES 
Obtain the following reduction formulas. 
1. 
2. 
3. 
4. 
5. 
Csin" 
/' 
_ , sin n+1 x cos m_1 x . m — 1 
x cos'" x clx = f- — 
m + n 
J sin" 
cos” z dx = + 1 
m -f- n 
cos™- 2 x dx. 
x cos'" -2 x dx. 
m 
sin" +1 X 
m 
m 
(m — 1) cos'" -1 x 
S’ 
-n — 2 Us in" x dx 
a — 1 J cos’" -2 x 
7)1 — 
Usin" xdx _ 
J COS'" X 
r dx _ _ 
J cos’"a; (m — 1)cos’" -1 ® ’ m — i j cos 7 " -2 » 
J*tan" x dx — —— — J*tan" -2 x dx. 
f/« 
6. Obtain the formula of Question 2 directly by starting with 
d(sin x cos' 1 x). 
7. Obtain the formula of Question 4 in a similar manner. 
8. Check the formulas of the Exercises against the corresponding 
formulas of the text by setting x = \ it — y. 
9. Obtain the formula of Question 5 by starting with d tali' 1 x and 
making a suitable trigonometric reduction of the result. 
10. 
Evaluate the following integrals: 
(a) 
J 
cos 4 a; dx. 
(b) J 
f cos 3 xdx. 
(e) 
j* sin x cos 2 x dx. 
(d) 
J 
Ç dx 
' cos 4 X 
W j 
f dx , ■ 
cos 3 # 
(f) 
Usin xdx 
J COS 2 X 
2. 
It 
para£ 
singl 
need 
(1) 
In 
Thu 
(2) 
(3)