Understanding Engineering Mathematics

Applying the result a number of times we see for example that

∫

3 f(x)dx=

∫

(f (x)+f(x)+f(x))dx

=

∫

f(x)dx+

∫

f(x)dx+

∫

f(x)dx

= 3

∫

f(x)dx

and in general for any numerical multiplierk:

∫

kf (x )dx=k

∫

f(x)dx

In general, ifkandlare constants then we have

∫

(kf+lg)dx=k

∫

fdx+l

∫

gdx

For this reason integration is called alinear operation. This rule enables us to integrate
any linear combination of standard integrals, including all polynomials, for example.

Solution to review question 9.1.3

The given function is a linear combination of standard integrals. We can

therefore use the linearity of the integral operation:

∫ (

2 x^2 +

3

x^2

+ 4

√

x− 2 ex+4sinx

)

dx

= 2

∫

x^2 dx+ 3

∫

dx

x^2

+ 4

∫

x^1 /^2 dx

− 2

∫

exdx+ 4

∫

sinxdx

=

2 x^3

3

−

3

x

+

8 x^3 /^2

3

− 2 ex−4cosx+C

Note that we only need one arbitrary constant for the overall integral and

not one for each of the ‘summands’.

9.2.4 Simplifying the integrand

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When faced with a new integral, the first thing to do, after checking whether it is a standard
integral, is to see if it is in the most convenient form for integration. Thus, in the integral∫

f(x)dx, we have two things to play with:

• the integrandf(x)


• the variablex