Rules of Integration
Table of Integrals
Z
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f (x)
For any function f and g and any constant value k:
Z
Multiplicative constant:
xn ,
Z
kf (x)dx = k
f (x)dx
Z
Z
(f (x) ± g(x))dx =
(ax + b)n
Z
f (x)dx ±
g(x)dx
f 0 (x)(f (x))n
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Integration for Economics and Business Studies
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Author: Morgiane Richard, University of Aberdeen
Z
f (x)g 0 (x)dx = f (x)g(x) −
Z
The definite integral is written:
Z
f (x)dx = F (x) + C with F 0 (x) = f (x)
eax+b
eax+b
+C
a
f 0 (x)ef (x)
ef (x) + C
1
x
1
ax + b
f 0 (x)
f (x)
Definite Integral
Indefinite Integral
b
f (x)dx and is equal to:
a
Z
b
f (x)dx =
[F (x)]ba
= F (b) − F (a)
1
(f (x))2 + C
2
1 (ax + b)n+1
+C
a
n+1
1
(f (x))n+1 + C
n+1
ex + C
f 0 (x)g(x)dx
Z
xn+1
+C
n+1
ex
Integration by part (to integrate products of functions):
Reviewer: Anthony Cronin, University College Dublin
Integration is the opposite of differentiation, and the indefinite integral of a function is the opposite of the derivative of a function.
The indefinite integral of a function f
Z
is written f (x)dx. For any function f :
n 6= −1
f 0 (x)f (x)
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Addition rule:
f (x)dx
ln x + C
ln (ax + b)
+C
a
ln (f (x)) + C
Table 2: Integral of functions commonly used in Economics and
Business Studies.
a
where F is a primitive of f . The definite integral is a measure of the area between the x-axis and the curve of f ,
between the points x = a and x = b.
f(x)=exp(2x)
8
Area between
the curve of f
and the x−axis
between the
points x=0 and
x=1
6
4
C is a constant of integration. The indefinite integral is
always defined with a constant of integration because all
functions that differ by a constant have the same derivative:
dF (x)
d(F (x) + C)
=
= f (x)
dx
dx
The functions F (x) + C are called primitives of the funtion
f.
x0.5
x
1/n
x
−n
= x1/2 =
=
=
xn+m
=
xn−m
=
√
x
√
n
x
1
xn
n m
x x
xn
xm
Table 1: Reminder on index laws.
1
2
0
0
0.5
1
1.5
Figure 1: The definite integral of a function is the area between
its graph and the x-axis between two specific values of x.
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