Some Formulae of Integrals
Basic integrals
Z
1
xn+1 + C, n 6= −1
n+1
Z
Z
1
1
1
dx = ln |x| + C
dx = ln |ax + b| + C
x
ax + b
a
Z
Z
1
ex dx = ex + C
eax dx = eax + C
a
xn dx =
Trig integrals
Z
Z
sin xdx = − cos x + C
Z
1
sin(ax)dx = − cos(ax) + C
a
cos xdx = sin x + C,
Z
cos(ax)dx =
Z
1
sin(ax) + C
a
2
Z
csc2 xdx = − cot x + C
Z
Z
sec x tan xdx = sec x + C
csc x cot xdx = − csc x + C
Z
Z
tan xdx = ln | sec x| + C
cot xdx = ln | sin x| + C
Z
Z
sec xdx = ln | sec x + tan x| + C
csc xdx = ln | csc x + cot x| + C
sec xdx = tan x + C
Fractions
x−a
1
1
ln |
|+C
dx =
2
−a
2a
x+a
Z
p
1
√
dx = ln |x ± x2 + a2 | + C
x2 ± a2
Z
Z
x
1
1
dx = tan−1 ( ) + C,
2
2
x +a
a
a
Z
x
1
√
dx = sin−1 ( ) + C
a
a2 − x2
x2
Hyperbolic functions
Z
sinh xdx = cosh x + C
Z
cosh xdx = sinh x + C
NOTE:
(1) The above are some integrals often used in MA307. You may find a more complete table if needed.
(2) For integrals which are not in this table, you may need some method of integrations from your calculus course. Basically, you should review (a) integration by parts; (b) integration by substitution;
(c) partial fractions.
(3) For integrals involving trig functions, you may need the trig identities. Please review them as well.
(4) Recall that the hyperbolic functions are defined as
1
1
sinh x = (ex − e−x ), cosh x = (ex + e−x )
2
2
1