(xiii) sin 2xcos 2x is, from the double angle formulae,^12 sin 4x and
becomes a standard integral on substitutingu= 4 x(271
➤
).(xiv)x− 1
√
x^2 − 2 x+ 3looks fearsome until we notice that the numerator
is almost the derivative of the quadratic under the square root,
prompting us to make the substitutionu=x^2 − 2 x+3 (263
➤
).(xv)x+ 3
x^2 + 2 x+ 2can be rewritten asx+ 1
x^2 + 2 x+ 2+2
x^2 + 2 x+ 2
whence the first fraction can be integrated by the substitutionu=
x^2 + 2 x+2, and the second by completing the square (268➤
).
(xvi) sin( 4 x)cos( 5 x)looks like (xiii) but in this case we have to use
the compound angle formulae to express it as a combination of
sin( 9 x)and sinx– a similar integral is done in Review Ques-
tion 9.1.8(iv) (271
➤
).
(xvii) lnxcos(x^2 + 1 )just notice that thexis almost the derivative of the
x^2 +1 and take it from there (263
➤
).9.2.12 The definite integral
➤
253 284➤If ∫
f(x)dx=F(x)+Cthen thedefinite integral off.x/between the limitsx=a,x=bis thenumber:
∫b
af(x)dx=[F(x)]ba=F(b)−F(a)That is, substitute the upper limitbin the indefinite integralF(x)and subtract the value of
F(x)with the lower limit substituted. Notice that even if we include the arbitrary constant
in the actual integration, it will only cancel out when the difference between the upper
and lower limits is taken – so we can discard it in the definite integral. The variablexin
the definite integral is called adummy variable, it being immaterial to the value of the
integral – thus
∫b
af(x)dx=∫baf(t)dtIn this respect the integration variable is like the summation index in the sigma notation
(102
➤
). As we will see in Chapter 10, the definite integral can be interpreted as an
area under a curve, and more rigorously as the limit of a sum – but here we are simply
interested in the technicalities ofperformingthe definite integral.
Obvious properties of the definite integral are:
∫a
af(x)dx=F(a)−F(a)= 0
∫baf(x)dx=F(b)−F(a)=−(F (a)−F(b))=−∫abf(x)dx