128 Chapter 5Integration
(5.2)
where Cis an arbitrary constant. For example, if y 1 = 1 x
2then 1 = 12 x, and the
indefinite integral of the function 2 xis
The symbol
Z
is called the integral sign; it is an elongated ‘S’ (for summation) and has
its origins in Leibniz’s formulation of the integral calculus; its significance will
become clearer in Section 5.4 when we discuss the integral as the limit of a sum. The
function to be integrated,F′(x)in (5.2), is called the integrand, xis the variable of
integrationand dxis called the element of x. Cis an arbitrary constant called the
integration constant. It is included as part of the value of the indefinite integral
because, giveny 1 = 1 F(x)with derivativeF′(x), the function (y 1 + 1 C) also has derivative
F′(x)
(5.3)
Table 5.1 is a short list of ‘standard integrals’ involving some of the more important
elementary functions (compare Table 4.2). Each entry in the list can be checked by
differentiation of the right side of the equation; for example,
so that
General methods of integration and further standard integrals are discussed in
Chapter 6. A more comprehensive list of standard integrals is given in the Appendix.
Z
11
ax b
dx
a
ax b C
=++ln( )
d
dx
ax b C
a
ax b
ln( ++)
=
d
dx
yC
dy
dx
dC
dx
dy
dx
()+= + =
Z 22
22x dx x C because
d
dx
=+ ()xC x+=
dy
dx
ZFxdx Fx C′() =+()
Table 5.1 Elementary integrals
- =
- =
- =
- =
- =
1
a
Z ln(ax b C++)
1
ax b
dx
1
a
Zcosax dx sinax C+
−+
1
a
Zsinax dx cosax C
1
a
eC
axZedx +
axx
a
Ca
a++≠−
1
1
Zxdx 1
a