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

440 Chapter 13 Integration
1
(r)
25+t 2
1
(s) √
25−t 2
(t)
6
1+x 2 + 1+x2
6
6 Theacceleration,a, ofaparticleis the rate ofchange
ofspeed, v,withrespecttotimet,thatisa = dv
dt .The
speed ofthe particle is the rate ofchange of
distance,s,that is v = ds .Ifthe acceleration isgiven
dt
by1+ t ,find expressionsforspeed anddistance.
2
7 Thespeed, v,ofaparticlevaries with time according
to
v(t) =2−e −t
(a) Obtainan expression forthe distancetravelled by
the particle.
(b) Calculate the distancetravelled bythe particle
betweent = 0 andt = 3.
8 Bywritingsinhax andcoshax in terms ofthe
exponential functionfind
∫
(a) sinhax dx
∫
(b) coshax dx
(c) Useyour results from (a)and(b)to find
∫
3sinh2x +cosh4x dx.
9 Acapacitor ofcapacitance10 −2 Fhas acurrenti(t)
through itwhere
i(t) =10−e −t
Find an expression forthe voltageacross the
capacitor.
10 Integratethe following:
1 1
(a)
x 2 (b)
+4 2x 2 +4
3
(c)
2x 2 +1
2
(e) √
4−x 2
1
(g) √
1 − (x 2 /2)
(d)
(f)
1
√
9−x 2
−7
√
2−3x 2
11 Bywriting 3 +x in the form 3 x x +1,
∫ 3 +x
find dx.
x
12 (a) Express
x 2 +2x+1
x(x 2 +1)
asitspartial fractions.
∫ x 2 +2x+1
(b) Hence find
x(x 2 dx.
+1)
13 Thevelocity, v,ofaparticleisgiven by
v=2+e −t/2
(a) Given distance,s,and v are relatedby ds
dt = v
findan expression fordistance.
(b) Acceleration isthe rate ofchange ofvelocity
with respect tot.Determine the acceleration.
14 (a) Usethe product ruleofdifferentiationto verify
d
dx (xe2x ) = e 2x +2xe 2x
(b) Hence show
∫
xe 2x dx = xe2x
2 − e2x
4 +c
15 Integrate
2−t 2
(a)
t 3
(c)
(e)
cos4x
sin4x
4+e 2t
(b)
e 3t
1
(d)
2sin3x
2+x 2
1+x 2 (f) sin 2 t +cos 2 t
Solutions
x 11
1 (a) +c (b) 9x+c
11
2 2
(c)
5 x2.5 +c (d)
3 t1.5 +c
(e) ln |z| +c (f) −3.2x +c
(g) − 1
2x 2 +c (h) 2√ t +c
x 7
(i)
7 +c (j) −1 x +c
3
(k)
4 t4/3 +c