The Chemistry Maths Book, Second Edition

5.2 The indefinite integral 129

EXAMPLES 5.1Indefinite integrals

(i)

(ii)

(iii)

(iv)

(v)

(vi)

We note that if we putC 1 = 1 ln 1 Ain (vi) then

0 Exercises 1–10

In every case, the effect of the operation

Z

...dx is to reverse the effect of

differentiation; the integral of the derivativeof a function retrieves the function.

Also, differentiating both sides of equation (5.2) gives

(5.4)

so that the derivative of the integralof a function retrieves that function.

Differential and integral operators

An alternative way of describing the operations of differentiation and integration,

that does not involve Leibniz’s symbolism, makes use of the differential operator

D 1 = 1 d 2 dxintroduced in Section 4.2, with property

DF(x) 1 = 1 F′(x)

The effect of DonF(x)is to transform it into its derivativeF′(x), and a corresponding

inverse operatorD

− 1

, an integral operator, can be defined whose effect is to reverse

that of D. Thus

D

− 1

F′(x) 1 = 1 F(x) 1 + 1 C

d

dx

Fxdx

d

dx

Z ′ =+Fx C F x













() () = ′()

Z

1

3

33

x

dx x A A x

• 
  

=++= +ln( ) ln ln ( )

Z

1

3

3

x

dx x C

• 
  

=++ln( )

Zcos 2θθdC=+sin θ

1

2

2

Zedt e C

22 tt

1

2

=+

ZZdx==+ 1 dx x C

ZZ

dx

x

xdx

x

C

x

== Cx

−+

+= +=

−

−+

12

12 1 12

12

12 1 12

2

()

()

++=CxC 2 +

ZZ

dx

x

xdx

x

C

x

C

x

C

2

2

21 1

21 1

1

==

−+

+=

−

+=−+

−

−+ −