The Chemistry Maths Book, Second Edition

5.2 The indefinite integral 131

EXAMPLE 5.2Find subject to conditions (i)(x, 1 y) 1 = 1 (2, 1 0), (ii)y 1 = 110

whenx 1 = 13.

We have. Then

(i) 0 1 = 12

2

1 + 1 C,C 1 = 1 −4,y 1 = 1 x

2

1 − 1 4 (ii) 10 1 = 13

2

1 + 1 C,C 1 = 1 1,y 1 = 1 x

2

1 + 11

0 Exercises 11, 12

Two rules

1.For a multiple of a function:

(5.6)

2.For a sum of functions:

(5.7)

It follows from these rules that the integral of a linear combination of functions,

f(x) 1 = 1 a 1 u(x) 1 + 1 bv(x) 1 + 1 cw(x) 1 +1- (5.8)

can be written as the sum of integrals

(5.9)

EXAMPLE 5.3Integral of a linear combination of functions

0 Exercises 13–15

=× + ×−

















−−

( )

+= −

−

3

4

2

1

3

3

3

4

2

3

4

4

x

xeCx

x

cos coos 3xe C

x

++

−

ZZ 323 3 2 3ZZ

33

x x e dx x dx x dx e dx

xx

+−

( )

=+ −

−−

sin sin

ZZZZfxdxauxdxb xdxc xdx() =++ +() vw() () 

ZZZux x dx ux dx() ()+ () ()x dx













vv=+

ZZau x dx a u x dx() = ()

yxdxxC==+Z 2

2

yxdx=Z 2