Understanding Engineering Mathematics

fromA(iii), we see that

d

dx

(

2

3

e^3 x

)

= 2 e^3 x

So, again remembering the arbitrary constant:

∫

2 e^3 xdx=

2

3

e^3 x+C

9.2.2 Standard integrals

➤

251 281➤

We can compile a useful list of integrals just by reading a table of derivatives backwards.
The simplest example is the derivative ofxα. For any powerα(that is,αcan be any
positive or negative real number – check that the result does make sense whenα=0)
we have

d

dx

(xα)=αxα−^1

from which, for any powerα,excepta=− 1 (see below)

d

dx

(xα+^1 )=(α+ 1 )xα

and so ∫

xαdx=

xα+^1
α+ 1
We must here exclude the caseα=−1 because otherwise we will have zero in the
denominator. In fact, the result forα=−1is:
∫
x−^1 dx=

∫

dx

x

=ln|x|+C

which we know from

d

dx

(ln|x|)=

1

x

.

Similarly, from the derivatives of other functions we can build up the table of standard
integrals given below (from which the arbitrary constant has been omitted). Note that much
of it is the table of standard derivatives given in Chapter 8 read ‘backwards’ (233
➤
). The
term ‘standard integral’ is of course relative – an advanced calculus book might have many
more such integrals that you are expected to know. Here however we confine ourselves
to the most important elementary functions. Obviously, the more you do know, the easier
integration will be, and in particular you will find it invaluable to commit all those in the
table to memory.
Much of integration involves manipulating the integrand, or the whole integral, to make
it expressible in terms of the standard integrals. Points to note about integration are:

• the better your differentiation, the better your integration


• know your basic algebra and trig skills


• practice and perseverance pay dividends


• trial and error may be needed