8.7
Table of Integrals
Engineers usually refer to a table of integrals when performing calculations involving integration.
This leaflet provides such a table. Sometimes restrictions need to be placed on the values of
some of the variables. These restrictions are shown in the third column.
1. A table of integrals
f (x)
k, any constant
x
x2
xn
x−1 = x1
ex
ekx
cos x
cos kx
sin x
sin kx
tan x
sec x
cosec x
cot x
cosh x
sinh x
tanh x
coth x
1
x2 +a2
1
x2 −a2
1
a2 −x2
√ 1
x2 +a2
√ 1
x2 −a2
√ 1
x2 +k
√ 1
a2 −x2
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R
f (x) dx
kx + c
x2
+c
2
x3
+c
3
xn+1
+c
n+1
ln |x| + c
ex + c
1 kx
e +c
k
sin x + c
1
sin kx + c
k
− cos x + c
− k1 cos kx + c
ln(sec x) + c
ln(sec x + tan x) + c
ln(cosec x−cot x) + c
ln(sin x) + c
sinh x + c
cosh x + c
ln cosh x + c
ln sinh x + c
1
tan−1 xa + c
a
1
ln x−a
+c
2a
x+a
1
a+x
ln a−x + c
2a
sinh−1 xa + c
cosh−1 xa + c
√
ln(x + x2 + k) + c
sin−1 xa + c
8.7.1
n 6= −1
− π2 < x < π2
− π2 < x < π2
0<x<π
0<x<π
x>0
a>0
|x| > a > 0
|x| < a
a>0
x>a>0
−a 6 x 6 a
c Pearson Education Ltd 2000
Exercises
1. In each case, use the Table of Integrals to integrate the given function with respect to x.
a) x
b) x6
c) x−2
d) x−3
e) x−1 , (be careful!)
f) x1/2
g) x−1/2
h) e3x
i) e7x
j) e−2x
k) e0.5x
l) ex
m) e−x
n) cos x
o) sin x
p) sin 3x
q) cos 2x
r) 5
2. You should be able to use the table when variables other than x are involved. Use the table
to integrate each of the following functions with respect to t.
√
a) et ,
b) e5t ,
c) t7 ,
d) t,
e) cos 5t,
f) e−t .
Answers
2
1. a) x2 + c, b)
− 2x12 + c,
e) ln |x| + c,
f)
x7
7
x3/2
3/2
+ c,
c)
+ c = −x−1 + c, or − x1 + c,
+ c = 32 x3/2 + c, g)
j) − 21 e−2x + c,
k) 2e0.5x + c,
2. a) et + c, b)
e5t
5
p) − 13 cos 3x + c,
x−1
−1
q)
1
2
+ c,
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+ c = 2x1/2 + c,
x−2
−2
t8
8
r) 5x + c.
+ c, d)
2t3/2
3
+ c, e)
8.7.2
sin 5t
5
+ c = − 12 x−2 + c, or
h) 31 e3x + c,
l) ex + c m) −e−x + c, n) sin x + c,
sin 2x + c,
c)
x1/2
1/2
d)
i) 71 e7x + c,
o) − cos x + c,
+ c, f) −e−t + c.
c Pearson Education Ltd 2000