Understanding Engineering Mathematics

Note that we only need the one arbitrary constant. We can now

transfer theIto the left-hand side to get

2 I=ex(sinx−cosx)+C

so I=^12 ex(sinx−cosx)+C

NB. SinceCis arbitrary we have no need to keep changing it during

the manipulations, or introducing new arbitrary constants.

9.2.11 Choice of integration methods

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So far you have usually been told which method to use for a particular integral – a bit
like someone handing you just the right tool for a particular job when you need it. By
now you have developed a whole box of tools of integration, and there will not always be
someone around to tell you which to choose for the given jobs – you need to know the
tools and their uses well enough to choose the right ones for yourself. Efficient selection
of the right tool from the box, as you will know, requires a lot of practice and experience,
and even the best of us will make mistakes at times. Sometimes the tool we pick won’t
fit, so we have to try another. Sometimes different tools will do the same job, but one of
them is the best to use in a particular instance. It helps to have a rough summary of the
different types of tools, and for integration we have covered essentially:

• standard integrals


• simplifying the integrand


• linear substitution


• du=f′(x)dxsubstitutions


• the use of trig identities and substitutions


• partial fractions


• completing the square


• integration by parts
  Various strategies for choosing the best methods are discussed in other sections, and
  here we will simply use the review question to bring together more examples. Be prepared
  to go up a few blind alleys and don’t be afraid to make mistakes in this topic.

Solution to review question 9.1.11

(i) We looked at rational functions such as

x− 3

x^2 − 6 x+ 4

in

Section 9.2.7, and the common methods are

• simplification of the integrand (dividing out, etc.)


• substitution,du=f′(x)dx


• partial fractions


• completing the square
  In this case we notice that the numerator is almost the derivative
  of the denominator, suggesting that we try the substitution u=
  x^2 − 6 x+4 here.