082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

434 Chapter 13 Integration
∫
(m)
∫
(n)
∫
(o)
∫
(p)
e t +e −t dt=e t −e −t +c
3sec(4x −1)dx = 3 ln|sec(4x −1) +tan(4x −1)| +c
4
2cot9xdx= 2 9 ln|sin9x|+c
7cosec(π/3) dx = {7cosec(π/3)}x +c
as cosec(π/3) isaconstant
Engineeringapplication13.1
Distancetravelledbyaparticle
The speed, v, of a particle is the rate of change of distance,s, with respect to timet,
thatis v = ds .Thespeedattimet isgivenby3+2t.ThisisillustratedinFigure13.1.
dt
Note that the speed of the particle is increasing linearly with time. Find the distance
intermsoft.
Solution
We aregiven that
v = ds
dt =3+2t
and arerequired tofinds. Therefore,
∫
s = 3+2tdt=3t+t 2 +c
Notethatthespeedoftheparticleismodelledbyalinearexpressionint.Thismeans
thatthespeedincreasesbythesameamountineachsubsequentsecond.Ontheother
hand,thedistance,s,ismodelledbyaquadraticint.Therefore,thedistancetravelled
ineachsubsequentsecondincreases.Figure13.1illustratesthespeed-timegraphand
Figure 13.2 illustratesthe distance-time graph.
y
s
3
c
O
Figure13.1
Graphofspeed ofthe particle, v,against
time,t.
t
O
Figure13.2
Graphofdistancetravelled by the
particle,s,against time,t.
t