250 INTEGRATION OF FUNCTIONS OF ONE VARIABLE, 
By simple algebraic transformations we may frequently 
put an integral into a shape in which one or other of the 
preceding formulas is at once applicable. 
. . rdx x n ~ l 1 rdx w6a?” _1 1 
(I) JTib* m TbJ a + b* ~^°g( a + b ^- 
+ bx r ‘ 
dx 
r dec r dx r d (a-x) 
J (2ax-x 2 )% J {a 2 —(a-a?) 8 }* J |a 2 — (a—a?) 2 }^ 
= cos' 
< 3 > /; 
dx 
loi 
i 
= vers' 
a a 
(a? 2 + 2aa?)i + x 4 a 
a 
)• 
d (x 2 ) . . x 2 
= A sin" 1 — 
* a 2 
(2 ax + x 2 )i 
r dx x 1 r ayx*) 
K) J (at-afi)* 2 J {a 4 -(a? 2 ) 2 }i 
d (x 2 ) I 
-t- a? 4 ^ J a i + (a? 2 ) 2 2 a* \a‘ 
dx x r d (a? 2 — a 2 )i 
(6) / 
{(a? 8 — a 2 ) (b 2 — ¿V s ) J2 J \b 2 ~ a 2 — (a? 2 — (f) j Ì 
. . (X 2 — Cp\ 2 
“ s,n Iftwj ’ 
/— 
J a 4 
dx 
dx 
hx + cx 2 c 
b\ 2 4>ac — h 2 ’ 
x + — 4 
2c 
4 c 2 
which is integrated by (c) or by (d) according as 4ac - b 2 > 0 
or < 0. Hence we have 
v r dx 
< 7 > hm 
i = Tk tan 
4- X 4- x 2 33 
/2a? 4 1\ 
V 3i j 
/• dx 1 I2x — 1 4- 5i\ 
J 1 4 a? - a? 2 ~ 5* ° 8 1.2a? - 1 -5‘J 
dx 
dx 
0°) f- 
2x + 2x 2 
dx 
4 3 a? 4 2 x 2 
= tan 1 (2 a? — 1). 
f 2a? 4 1 Ì