Table of Integrals

Basic Formulas
K
d2
Ha
1.
2.
3.
4.
5.
6.
7.
14.
15.
17.
18.
20.
22.
23.
24.
1
x
ln b ab
sin ax dx =
− cos ax
a
cos ax dx =
sin ax
a
tan ax dx =
ln|cos ax|
a
u dv = uv −
v du
10. sin(2x) = 2 sin(x) cos(x)
11. cos(2x) = 2 cos2 (x) − 1
12. tan(x) =
1−cos(2x)
sin(2x)
13. a sin(x) + b cos(x) =
√
a2 + b2 sin(x + tan−1 ab )
Rational Functions
x−a x
x
b
dx
1
1
26.
dx
=
−
ln
|ax
+
b|
=
+
ln
ax+b
a
a2
x2 (x2 −a2 )
a2 x
2a3
x+a
x+a x2
2
2
2
x
x
1
x
bx
3b
dx
=
−
ln
27.
dx
=
−
−
+
2
2
2
2
2
2
3
(x −a )
2(a −x )
4a
x−a
ax+b
2a
a
a
dx
28. x2 +ax+b =
b2
a3 ln |ax + b|
⎧
dx
−1
2
2x+a
⎪
−1
√
√
⎪
tan
if 4b > a2
(ax+b)2 = a(ax+b)
⎪
4b−a2
⎨ 4b−a2
x
−1
b
1
if 4b = a2
x+a/2
(ax+b)2 dx = a2 (ax+b) + a2 ln |ax + b|
⎪
⎪
2
2x+a−√
⎪
1
√a −4b if 4b < a2
√
⎩
x2
ln
2
2
2
ax+b
b
a −4b
2x+a+ a −4b
(ax+b)2 dx = a3 − a3 (ax+b) −
2
x
1
+
2b
x
29.
dx
=
ln
+
ax
+
b
x2 +ax+b
2
a3 ln |ax + b|
x dx
⎧
1
a
2x+a
=
ln
−1
√
√
⎪
x(ax+b)
b
ax+b
tan
if 4b > a2
⎪
4b−a2
4b−a2
⎨
ax+b 1
dx
a
a
−
if 4b = a2
=
ln
2x+a
x2 (ax+b)
b2
x
bx
⎪
√
⎪
⎩ √−a ln 2x+a−√a2 −4b if 4b < a2
ax+b dx
2a
2ax+b
2 a2 −4b
2x+a+ a2 −4b
− b2 x(ax+b)
x2 (ax+b)2 = b3 ln
x
2 dx
2
1
x
x2
a
+
=
ln
2
2
2
2
2
30.
dx
=
x
−
ln
x
+
ax
+
b
x(x +a )
2a
x +a
x2 +ax+b
2
⎧
dx
−1
1
−1 x
2
=
−
tan
2x+a
a
−2b
−1
⎪
2
2
2
2
3
√
√
x (x +a )
a x
a
a
if 4b > a2
tan
⎪
⎪
4b−a2
4b−a2
⎪
⎨
x2
1
x
−1 x
−a2
−
dx
=
tan
if 4b = a2
(x2 +a2 )2
2a
a
2(x2 +a2 )
4x+2a
⎪
⎪
⎪
dx
2
2
2x+a−√
⎪
a2 −4b a2 −2b
1
x −a
⎩
√
√
ln 2x+a+ a2 −4b if 4b < a2
x(x2 −a2 ) = 2a2 ln x2 2 a2 −4b
1
m
25.
abx dx =
9. sin2 (x) = 1 − cos2 (x)
co
21.
= a1 ln |ax + b|
8. sin(x) = cos(x − π2 )
w.
19.
dx
ax+b
a
n+1
n+1 x
no
16.
axn dx =
31.
x2 +2ax+a2 =
ln x2 −ax+a2 +
1
6a2
33.
K
d2
Ha
dx
x3 +a3
√1
3a2
32.
tan
x
x3 +a3
√1
3a
tan
2x−a
√
3a
−1
34.
35.
2x−a
√
3a
=
dx
x4 +a4
√1
2 2a3
x2 −ax+a2 1
dx = 6a ln x2 +2ax+a2 +
−1
tan
x
x4 +a4
√1
4 2a3
−1
dx =
x2
x4 +a4
√
2 √
x +√2ax+a2 ln x2 − 2ax+a2 +
2a
a2 −x2
−1
2a2
tan
−1
2
a
x2
2 √
x −√2ax+a2 ln x2 + 2ax+a2 +
√1
4 2a
dx =
√
2a
√1 tan−1
a2 −x2
2 2a
Square Roots
√
2
ax + b dx = 3a
(ax + b)3/2
√
dx
37. √ax+b
= a2 ax + b
√
3/2
38. x ax + b dx = 6ax−4b
15a2 (ax + b)
√
x
39. √ax+b
dx = 2ax−4b
ax + b
2
3a
36.
ax+b
x
=
√dx
x ax+b
44.
47.
√
√ x
a x +b
=
√
2 x
a
dx =
x
a
−
−
2b
a2
+
2b2 √
x
a3
−
ln |a x + b|
2b x
a2
+
2
2b
a3
√
x2 + a2 dx = 12 x x2 + a2 +
√
ln x + x2 + a2 √
√
48.
x2 − a2 dx = 12 x x2 − a2 −
a2
2
√
ln x + x2 − a2 √
√
a2 − x2 dx = 12 x a2 − x2 +
√
50. x2 x2 + a2 dx =
49.
√
√
√
a2
2
2(as−br) u2
dx
=
2
2
rx+s
r
(u −a/c)2 du
where u = ax+b
rx+s , see eqs. 24 & 27
√dx
a x +b
bx
a2
2x3 +a2 x
8
51.
ln |a x + b|
2
x2 + a2 −
√
x2 x2 − a2 dx =
2x3 −a2 x
8
√
√
√
a4
8
a2
2
sin−1 xa √
ln x + x2 + a2 √
ln x + x2 − a2 m
43.
−
√
ln |a x + b|
√
46.
x + a x + b dx =
√
2
a√
2b
a2
x
−
x
+
−
x + b)3/2 −
(x
+
a
3
6
3
4
√
√
a3
a
ab − 4 ln x + 2 + x + a x + b
ax+b
2 3/2
3a x
co
⎧
√
√ ax+b
−
1
⎪
⎨ √b ln √ax+b +√bb if b > 0
⎪
ax+b
⎩ √2 tan−1
if b < 0
−b
−b
42.
dx =
w.
41.
√
√x
a x +b
2b3
a4
√
dx = 2 ax + b +
⎧√
√
√ax+b −√
b
⎪
⎨ b ln ax+b +√b if b > 0
√
⎪
ax+b
⎩−2 −b tan−1
if b < 0
−b
no
40.
45.
x 2 − a2 −
a4
8
52.
√
x2 a2 − x2 dx =
x
K
d2
Ha
√
63.
2x3 −a2 x
8
√x2 +a2
x
4
a2 − x2 + a8 sin−1
√
53.
dx = x2 + a2 −
√
a+ x2 +a2 a ln x
54.
55.
56.
√
√
√
x2 −a2
x
a2 −x2
x
dx =
dx =
√
a+ a2 −x2 a ln x
x2 +a2
x2
√
√
dx = −
x
a2 − x 2 −
√
x2 +a2
x
+
x2 −a2
x
+
√
ln x + x2 + a2 √
x2 −a2
x2
dx = −
√
√
ln x + x2 − a2 58.
59.
61.
62.
a
2
a2 −x2
x2
√ dx
x2 +a2
√ dx
x2 −a2
√ dx
a2 −x2
2
√ x
2
x +a2
√
2
2
dx = − a x−x − sin−1
√
= ln x + x2 + a2 √
= ln x + x2 − a2 = sin−1 xa
√
dx = 12 x x2 + a2 −
√
ln x + x2 + a2 √
ln x + x2 − a2 √
dx = 12 x a2 − x2 +
√
a+ x2 +a2 dx
−1
√
65. x x2 +a2 = a ln x
66. x√xdx2 −a2 = a1 sec−1 xa √
2 −x2 a+
a
dx
−1
67. x√a2 −x2 = a ln x
√
− x2 +a2
dx
√
68. x2 x2 +a2 = a2 x
√
2
2
dx
√
69. x2 x2 −a2 = xa2−a
x
√
− a2 −x2
dx
√
70. x2 a2 −x2 = a2 x
√
x
2
2
71. x+√dx
=
2a2 x + a −
x2 +a2
64.
2
√ x
2
a −x2
x2
2a2
x
a
72.
+ 12 ln(x +
√dx
x+ x2 −a2
x2
2a2
a2
2
sin−1
x
a
√
x 2 + a2 )
√
−x
= 2a
x 2 − a2 +
2
+ 12 ln(x +
√
x2 − a2 )
√
73. x+√dx
=
ln(x
+
a2 − x2 ) −
a2 −x2
1
1
2
2
−1 √ x
4 ln |2x − a | + 2 tan
a2 −x2
74.
√dx
a+ a2 −x2
tan
−1
=
√ x
a2 −x2
√
a2 −x2
x
−
+
a
x
co
2
√
√
dx = 12 x x2 − a2 +
w.
60.
a
2
√ x
2
x −a2
no
57.
a2
2
a
x2 − a2 − a sec−1
3
m
Natural Logarithms
dx
n+1
xn+1
78.
,
n
=
1
75. xn ln x dx = x n+1ln x − (n+1)
2
x ln x = ln |ln x|
x
ln x
1
ln x
79.
dx
=
ln
x
−
ln
|x
+
a|
1
2
(x+a)2
a x+a
76.
x dx = 2 (ln x)
80.
(ln x)2 dx = x[(ln x)2 − 2 ln x + 2]
(ln x)m
m+1
x)
77.
dx = (lnm+1
, m = −1
x
81.
√
ln(a x + b) dx =
85.
K
d2
Ha
83.
ln(x +
+
x2
a2 )
dx = −
√
x2
+
√
a2
87.
x 2 − a2 )
ln(a +
x ln(a +
+
88.
√
√
a2 − x2 ) dx = x +
a2 − x2 ) − a sin−1
x
a
ln(x3 + a3 ) dx = x ln(x3 + a3 ) − x +
x2 +2ax+a2 √
a
−1 √2x
3a tan
2 ln x2 −ax+a2 +
3a
no
√
x2 − a2 ) dx = 2x ln a +
√
x ln(x + a2 − x2 ) − x −
√
a+ a2 −x2 a
2 ln x
x ln(x + x2 + a2 )
√
√
84. ln(x + x2 − a2 ) dx = − x2 − a2 +
x ln(x +
√
√
x2 − a2 − x ln(x + x2 − a2 )
√
86. ln(x + a2 − x2 ) dx =
(x + a2 ) ln(x2 + ax + b)+
⎧√
−1 √ 2x
2
⎪
if 4b > a2
4b − a tan
⎪
2
⎪
4b−a
⎪
⎨
0
if 4b = a2
⎪
⎪
√
√
⎪
2 −4b ⎪
2x+a+
a
1
⎩ a2 − 4b ln √
if 4b < a2
2
2x+a− a2 −4b
√
ln(x −
√
√
√
2
(x − ab 2 ) ln(a x + b) − x2 + b a x
82. ln(x2 + ax + b) dx = −2x − a +
Exponential Functions
89.
91.
92.
xeax dx =
xeax
a
x2 eax dx =
ea
√
x
dx
b+eax
dx =
=
x
b
−
2 ax
x e
a
2√
a
−
−
xea
1
ab
eax
a2
96.
ax
√
2xe
a2
x
−
2e
a3
√
a x
+
2
a2 e
ax
ln |b + eax |
eax
b+eax
95.
√
a
b − eax +
=
√
b
a
√
√b−eax −√
b
ln b−eax +√b 97.
√ dx
eax +b
=
⎧√
√
√eax +b −√
b
b
⎪
√
⎨ a ln eax +b + b if b > 0
√
√
⎪
⎩ 2 −b tan−1 √eax +b
if b < 0
a
−b
98.
eax sin(bx) dx =
eax
a2 +b2 [a sin(bx)
99.
eax cos(bx) dx =
eax
a2 +b2
4
− b cos(bx)]
m
b − eax dx =
√
√
√eax +b−√
b
b
√
a ln eax +b+ b √
2
√ dx
b−eax
co
dx = a1 ln |b + eax |
√
√
eax + b dx = a2 eax + b +
94.
⎧√
√
√eax +b −√
b
b
⎪
⎨ a ln eax +b +√b if b > 0
√
√
⎪
⎩− 2 −b tan−1 √eax +b
if b < 0
a
−b
93.
w.
90.
[b sin(bx) + a cos(bx)]
100.
xeax sin(bx) dx =
eax
a2 +b2
[ax sin(bx) −
K
d2
Ha
bx cos(bx) −
a2 −b2
a2 +b2
sin(bx) +
2ab
a2 +b2
101.
xeax cos(bx) dx =
ax cos(bx) −
cos(bx)]
a2 −b2
a2 +b2
eax
a2 +b2
[bx sin(bx) +
cos(bx) −
2ab
a2 +b2
sin(bx)]
Trigonometric Functions
102.
103.
2
3
104.
105.
106.
107.
108.
x
2
−
sin(2x)
4
2
sin3 (x) dx = − sin (x) cos(x) −
cos3 (x)
cos2 (x) dx =
x
2
+
sin(2x)
4
3
2
cos (x) dx = sin(x) cos (x) +
sin3 (x)
sin2 (x) cos2 (x) dx =
x
16
−
sin(4x)
32
x sin(x) dx = sin(x) − x cos(x)
x2 sin(x) dx = 2 cos(x) +
112.
117.
118.
sin(2x)
8
x cos(x) dx = cos(x) + x sin(x)
2
x cos(x) dx = −2 sin(x) +
sin(2x)
8
dx
cos(x)
dx
sin2 (x)
= − cot(x)
dx
cos2 (x)
= tan(x)
dx
sin3 (x)
− cos2 (x)
2 sin(x)
1−cos(x) 119.
=
+ ln sin(x) dx
1+sin(x) sin2 (x)
1
120. cos3 (x) = 2 cos(x) + 2 ln cos(x) 1+sin(x) dx
1
121. sin2 (x) cos(x) = ln cos(x) − sin(x)
1−cos(x) dx
1
122. sin(x) cos2 (x) = ln sin(x) + cos(x)
123. sin(x) sin(x + a) dx = 12 x cos(a) −
1
2
1
4
124.
sin(2x + a)
sin(ax) sin(bx) dx =
sin((a−b)x)
2(a−b)
−
sin((a+b)x)
2(a+b)
125.
sin((a−b)x)
2(a−b)
cos(ax) cos(bx) dx =
+
sin((a+b)x)
2(a+b)
126.
sin(ax) cos(bx) dx =
cos((a+b)x)
2(a+b)
127.
5
dx
1±sin(x)
− cos((a−b)x)
2(a−b)
−
m
2x cos(x) + x sin(x)
2
113. x cos2 (x) dx = x4 + x sin(2x)
+ cos(2x)
4
8
2
2
3
114. x cos2 (x) dx = x6 + x sin(2x)
+
4
−
2
x cos(2x)
4
co
111.
+
116.
1−cos(x) = ln sin(x) 1+sin(x) = ln cos(x) dx
sin(x)
w.
2x sin(x) − x2 cos(x)
2
109. x sin2 (x) dx = x4 − x sin(2x)
− cos(2x)
4
8
2 2
2
3
110. x sin (x) dx = x6 − x sin(2x)
−
4
x cos(2x)
4
115.
no
2
3
sin2 (x) dx =
= ∓ tan
π
4
∓
x
2
128.
= ∓ tan
π
4
∓
π
2x+π 130.
4
129.
x
1±sin(x)
dx = x tan
∓
4
x
2
∓
2 ln sin π4 ± x2 131.
dx
a+sin(x)
=
dx
a+cos(x)
=
x
1±cos(x)
dx = (x + π2 ) tan
π
4
∓
2x+π 4
π 2x+π 2 ln sin 4 ± 4
K
d2
Ha
dx
1±cos(x)
⎧
√
2x+π a2 −1
2
−1
⎪
√
⎨ a2 −1 tan
if |a| > 1
|a−1| tan
4
√
2
⎪
⎩ √ 1 2 ln a tan(x/2)+1−√1−a2 if |a| < 1
1−a
a tan(x/2)+1+ 1−a
132.
⎧
x |a−1|
2
−1
⎪
√
tan 2
if |a| > 1
⎪
⎨ √a2 −1 tan
a2 −1
√
1−a2 tan(x/2)+
⎪
1
⎪
√1−a if |a| < 1
⎩ √1−a2 ln 2 tan(x/2)− 1−a
1−a
134.
135.
137.
=
cos(x)
a sin(x)+cos(x) dx
√ 1
a2 +1
=
x+tan−1 a1 ln tan
2
x
a2 +1
+
a
2(a2 +1)
ln a +
tan2 (x) dx = tan(x) − x
tan3 (x) dx = 12 tan2 (x) + ln | cos(x)|
cot2 (x) dx = − cot(x) − x
sin(2x) +
143.
(cos−1 (x))2 dx = x(cos−1 (x))2 −
√
ln(x + x2 − 1)
145. (sin−1 (x))(cos−1 (x)) dx =
x sin−1 (x) cos−1 (x) + 2x +
√
146.
1 − x2 (cos−1 (x) − sin−1 (x))
sin(a sin−1 (x))dx =
m
− 1)
ln(x +
√
142. cos−1 (x) dx = x cos−1 (x) − 1 − x2
x2
a tan(x)+1 ln a tan(x)+a2 √
2x − 2 cos−1 (x) 1 − x2
144. cos−1 x1 dx = x cos−1 x1 −
√
2x + 2 sin−1 (x) 1 − x2
141. sin−1 x1 dx = x sin−1 x1 +
√
a
a2 +1
co
cot3 (x) dx = − 12 cot2 (x) − ln | sin(x)|
√
139. sin−1 (x) dx = x sin−1 (x) + 1 − x2
140. (sin−1 (x))2 dx = x(sin−1 (x))2 −
138.
a2 +1
2
w.
136.
dx
a sin(x)+cos(x)
no
133.
√
a 1−x2 cos(a sin−1 (x))+x sin(a sin−1 (x))
1−a2
6
∓
147.
cos(a cos−1 (x))dx =
149.
K
d2
Ha
√
a 1−x2 sin(a cos−1 (x))+x cos(a cos−1 (x))
1−a2
148.
1
2
153.
tan−1 (x) dx = x tan−1 (x) −
2
ln(x + 1)
tan−1
ax+b x+c
dx = x +
ex −e−x
2
155. cosh(x) =
ex +e−x
2
156. tanh(x) =
ex −e−x
ex +e−x
tan−1
tan(a + tan−1 (x)) dx =
x cot(a) −
150.
151.
152.
ax+b x+c
+
1
sin2 (a)
ln(x sin(a) − cos(a))
√
(2x2 −1) sin−1 x+x 1−x2
4
√
(2x2 −1) cos−1 x−x 1−x2
4
−1
x sin (x) dx =
x cos−1 (x) dx =
(x2 +1) tan−1 (x)−x
2
x tan−1 (x) dx =
b−ac
2a2 +2
ln((ax + b)2 + (x + c)2 )
Hyperbolic Functions
169. x sinh(x) dx = x cosh(x) − sinh(x)
170. x cosh(x) dx = x sinh(x) − cosh(x)
171. sinh−1 (x) dx = x sinh−1 (x) −
no
154. sinh(x) =
ab+c
a2 +1
√
172.
√
173.
x2 + 1
cosh−1 (x) dx = x cosh−1 (x) −
x2 − 1
tanh−1 (x) dx = 12 ln(1 − x2 ) +
w.
x tanh−1 (x)
174. x sinh−1 (x) dx =
2x2 +1
4
175.
√
x2 + 1
cosh−1 (x) −
1
4
√
x2 − 1
x tanh−1 (x) dx = 12 (x2 − 1) tanh−1 (x) +
m
7
1
4
x cosh−1 (x) dx =
2x2 +1
4
176.
sinh−1 (x) −
co
157. cosh2 (x) − sinh2 (x) = 1
√
158. sinh−1 = ln(x + x2 + 1)
√
159. cosh−1 = ln(x + x2 − 1)
−1
160. tanh = ln 1+y
1−y
161. sinh(x) dx = cosh(x)
162. cosh(x) dx = sinh(x)
163. tanh(x) dx = ln(ex + e−x )
x dx
= ln eex −1
164. sinh(x)
+1
dx
= 2 tan−1 (ex )
165. cosh(x)
−1
166. sinhdx2 (x) = − coth(x) = tanh(x)
167. coshdx2 (x) = tanh(x)
168. tanh2 (x) dx = x − tanh(x)
x
2
√
(sinh−1 (x))2 dx = 2x + x(sinh−1 (x))2 − 2 sinh−1 (x) x2 + 1
√
178. (cosh−1 (x))2 dx = 2x + x(cosh−1 (x))2 − 2 cosh−1 (x) x2 − 1
179. sinh−1 x1 dx = x sinh−1 x1 + sinh−1 (x)
−1 1
−1 1
−1 √ x
180. cosh
x dx = x cosh
x + tan
1−x2
177.
m
co
w.
no
K
d2
Ha
8