The Chemistry Maths Book, Second Edition

166 Chapter 6Methods of integration

f(x) 1 = 1 (2x 1 − 1 1)

3

1 = 18 x

3

1 − 112 x

2

1 + 16 x 1 − 11

= 12 x

4

1 − 14 x

3

1 + 13 x

2

1 − 1 x 1 + 1 C′

(C 1 = 1 C′ 1 − 1128 is an arbitrary constant).

A simpler way of integrating the function is to make the substitution

u 1 = 12 x 1 − 1 1,

where duis the differential ofu(x). Then, and

The integral has been transformed into a ‘standard integral’ by changing the variable

of integration from xto u. The method of substitution is also called integration by

change of variable.

In the general case, given the integral of a functionf(x)whose form is non-standard,

the method of substitution is to find a new variable u(x)such that

(6.7)

where the integral on the right is a standard integral; that is,g(u)is easier to integrate

thanf(x). Differentiating both sides of (6.7) with respect to xgives:

on the left side, by definition of the indefinite integral,

on the right side, by application of the chain rule,

Therefore

fx gu (6.8)

du

dx

() ()=

d

dx

gu du

d

du

gu du

du

dx

ZZ() ()

















=

















×=ggu

du

dx

()

d

dx

Zfxdx fx() ()

















=

ZZf x dx g u du() = ()

ZZ() 21 ()

1

2

1

8

1

8

21

334 4

xdx uduuC x C−= =+=−+

dx du=

1

2

du

du

dx

==dx dx 2

=−+

1

8

21

4

()xC

ZZZZZ()21 8 12 6

33 2

x−=dx x dx−x dx+ −xdx dx