Understanding Engineering Mathematics

9.1.11 Choice of integration methods ➤276 284➤➤

Discuss the methods you would use to integrate the following – you will be asked to
integrate them in RE9.3.11C.

(i)

x− 3

x^2 − 6 x+ 4

(ii) sin^3 x (iii)

elnx

x

(iv)

x− 1

2 x^2 +x− 3

(v) xe^3 x

2

(vi)

3

√

3 − 2 x−x^2

(vii) xsin(x+ 1 ) (viii) cos^4 x (ix) ln

(

ex

x

)

(x)

x+ 2

x^2 − 5 x+ 6

(xi) xe^2 x (xii) excos(ex)

(xiii) sin 2xcos 2x (xiv)

x− 1

√

x^2 − 2 x+ 3

(xv)

x+ 3

x^2 + 2 x+ 2

(xvi) sin 4xcos 5x (xvii) xcos(x^2 + 1 )

9.1.12 The definite integral ➤278 284➤➤

A.Evaluate

(i)

∫ 1

0

(x^2 + 1 )dx (ii)

∫ 1

0

xexdx (iii)

∫ 2

0

x^2

x^3 + 1

dx

(iv)

∫ 2 π

0

cosxsinxdx

B. What is wrong with

∫ 2

0

dx

x^2 − 1

?

9.2 Revision

9.2.1 Definition of integration

➤

251 280➤

Integration (or more strictly indefinite integration) is the reverse of differentiation.
Thus, if

dy

dx

=f(x)

then

y=

∫

f(x)dx

is theanti-derivativeorindefinite integraloff(x)(also sometimes called the ‘primi-
tive’).f(x)is called theintegrand.