Integration using trigonometric and hyperbolic substitutions 399
Table 40.1 Integrals using trigonometric and hyperbolic substitutions
f (x)
f (x)dx
Method
See problem
1. cos 2 x
1
2
x +
sin 2x
2
+ c
Use cos 2x = 2 cos 2 x − 1
1
2. sin 2 x
1
2
x −
sin 2x
2
+ c
Use cos 2x = 1 − 2 sin 2 x
2
3. tan 2 x
tan x − x + c
Use 1 + tan 2 x = sec 2 x
3
4. cot 2 x
− cot x − x + c
Use cot 2 x + 1 = cosec 2 x
4
5. cos m x sin n x (a) If either m or n is odd (but not both), use
cos 2 x + sin 2 x = 1
5, 6
(b) If both m and n are even, use either
cos 2x = 2 cos 2 x − 1 or cos 2x = 1 − 2 sin
2 x
7, 8
6. sin A cos B
Use
1
2 [ sin(A + B) + sin(A − B)]
9
7. cos A sin B
Use
1
2 [ sin(A + B) − sin(A − B)]
10
8. cos A cos B
Use
1
2 [ cos(A + B) + cos(A − B)]
11
9. sin A sin B
Use −
1
2 [ cos(A + B) − cos(A − B)]
12
10.
1
(a 2 − x 2 )
sin −1 x
a
+ c
Use x = a sin θ substitution
13, 14
11.
(a 2 − x 2 )
a 2
2
sin −1 x
a
+
x
2
(a 2 − x 2 ) + c
Use x = a sin θ substitution
15, 16
12.
1
a 2 + x 2
1
a
tan −1 x
a
+ c
Use x = a tan θ substitution
17–19
13.
1
(x 2 + a 2 )
sinh −1 x
a
+ c
Use x = a sinh θ substitution
20–22
or ln
x +
(x 2 + a 2 )
a
+ c
14.
(x 2 + a 2 )
a 2
2
sinh −1 x
a
+
x
2
(x 2 + a 2 ) + c
Use x = a sinh θ substitution
23
15.
1
(x 2 − a 2 )
cosh −1 x
a
+ c
Use x = a cosh θ substitution
24, 25
or ln
x +
(x 2 − a 2 )
a
+ c
16.
(x 2 − a 2 )
x
2
(x 2 − a 2 ) −
a 2
2
cosh −1 x
a
+ c Use x = a cosh θ substitution
26, 27