∫ dx
x√
x^2 n−a^2 n=−na^1 nsin−^1(
an
xn)
+C∫ √x+a
x−adx=√
x^2 −a^2 +acosh−^1 xa+C∫ √a+x
a−xdx=asin− 1 x
a−√
a^2 −x^2 +C∫
x√
a−x
a+xdx=a^2
2 cos− 1 (x
a)
+(x− 22 a)√
a^2 −x^2 +C, a > x∫
x√
a+x
a−xdx=a^2
2 sin− 1 x
a−x+ 2a
2√
a^2 −x^2 +C∫
(x+a)√
x+b
x−bdx= (x+a+b)√
x^2 −b^2 +b 2 (2a+b) cosh−^1 xb+C∫ dx
√
2 ax+x^2= ln|x+a+√
2 ax+x^2 |+C∫ √
ax^2 +c dx=
1
2 x√
ax^2 +c+ 2 √caln|√
ax+√
ax^2 +c|+c, a > 0
1
2 x√
ax^2 +c+ 2 √c−asin−^1(√−a
c x)
+C, a < 0∫ √1 +ax
1 −axdx=1
asin− (^1) x−^1
a
√
1 −x^2 +C
362.
∫ dx
(ax+b)^2 + (cx+d)^2 =
1
ad−bctan
− 1
[
(a^2 +c^2 )x+ (ab+cd)
ad−bc
]
+C, ad−bc 6 = 0
363.
∫ dx
(ax+b)^2 −(cx+d)^2 =
1
2(bc−ad)ln
∣∣
∣∣(a+c)x+ (b+d)
(a−c)x+ (b−d)
∣∣
∣∣+C, ad−bc 6 = 0
364.
∫ x dx
(ax^2 +b)^2 + (cx^2 +d)^2 =
1
2(ad−bc)tan
− 1
[(a (^2) +c (^2) )x (^2) + (ab+cd)
ad−bc
]
+C, ad−bc 6 = 0
365.
∫ dx
(x^2 +a^2 )(x^2 +b^2 )=
1
b^2 −a^2
( 1
atan
− 1 x
a−
1
btan
− 1 x
b
)
+C
366.
∫ (x (^2) +a (^2) )(x (^2) +b (^2) )
(x^2 +c^2 )(x^2 +d^2 )dx=x+
1
d^2 −c^2
[(a (^2) −c (^2) )(b (^2) −c (^2) )
c tan
− 1 x
c−
(a^2 −d^2 )(b^2 −d^2 )
d tan
− 1 x
d
]
+C
367.
∫ ax (^2) +b
(cx^2 +d)(ex^2 +f)dx=
√^1
cd
(ad−bc
ed−fc
)
tan−^1
(√c
dx
)
+√^1 ef
(af−be
fc−ed
)
tan−^1
(√e
fx
)
+C
368.
∫ x dx
(ax^2 +bx+c)^2 + (ax^2 −bx+c)^2 =
1
4 b
√
b^2 + 4ac
ln
∣∣
∣∣
∣
2 a^2 x^2 + 2ac+b^2 −b
√
b^2 + 4ac
2 a^2 x^2 + 2ac+b^2 +b
√
b^2 + 4ac
∣∣
∣∣
∣+C, b
(^2) + 4ac > 0
Appendix C