�
�
�
�
�
�
2
3
4
x
x
x
x
x
e = 1 + +
+
+
+ · · ·
1!
2!
3!
4!
3
5
7
x
x
x
sin x = x −
+
−
+ · · ·
3!
5!
7!
2
4
6
8
x
x
x
x
cos x = 1 −
+
−
+
− · · ·
2!
4!
6!
8!
3
5
7
x
x
x
sinh x = x +
+
+
+ · · ·
3!
5!
7!
2
4
6
x
x
x
cosh x = 1 +
+
+
+ · · ·
2!
4!
6!
−x
e x − e
sinh x =
2
−x
e x + e
cosh x =
2

d sin x = cos x dx; d cos x = − sin x dx

d sinh x = cosh x dx; d cosh x = + sinh x dx

sin x dx = − cos x + c;
cos x dx = sin x + c
1
1
1
1
sin
2 x dx = − sin x cos x + x + c = − sin 2 x + x + c
2
2
4
2
1
1
1
1
2
cos x dx = sin x cos x + x + c = sin 2x + x + c
2
2
4
2
sinh x dx = cosh x + c;
cosh x dx = sinh x + c
Appendix A: Mathematical Relations
and Functions
A.1 Useful Formulas
297