160 Chapter 5Integration
5.9 Exercises
Section 5.2
Evaluate the indefinite integrals:
Evaluate the indefinite integrals subject to the given conditions:
- ; y 1 = 1 0 whenx 1 = 13 12. ; y 1 = 1 0 whenx 1 = 1 π 24
- ;I 1 = 1 4whenx 1 = 12 14. ;I 1 = 1 3whenx 1 = 11
- ; I 1 = 1 0 whenx 1 = 10
Section 5.3
Evaluate the definite integrals:
- (i)Show that the rate equation of a first-order decomposition reaction
can be written in the logarithmic form
(ii)Integrate this equation with respect totover the range 0 to t, and show that
25.The Clausius–Clapeyron equation for liquid–vapour equilibrium is
If the enthalpy of vaporization, ∆H
vap, is constant in the temperature rangeT
1
toT
2
show, by integrating both sides of the equation with respect to T, that
wherep
1
1 = 1 p(T
1
)andp
2
1 = 1 p(T
2
).
ln
p
p
H
RTT
2
112
11
=
∆
−
vapdp
dT
H
RT
ln
=
∆
vap2
ln
()
()
() ( )
xt
x
kt x t x e
kt0
0
=− =
−
and
dx
dt
k
ln
=−
dx
dt
=−kx
Z
−π 40
Z sin 2 xdx
0
cos 3
πZ θθd
0
π 2Z cosθθd
1
5
3
edt
−tZ
2
3
2
dv
v+
Z
1
2
3
du
u
Z
3
5
Z dx
−
+
++
1
1
2
() 234 xxdx
Ixedx
x=−+ −
Z 44 2
1
2
2
cos
I
xx
x
= dx
++
Z
321
2
2
Ixxdx=++Z()523
4
yxd=Zcos4 x
yxdx=Z
2
Z
dx
3 −x
Z
dx
x− 1
Zedx
− 2 xZedx
3 xZsin 4xdx
Zxdx
− 13
Z
dx
x
3
Zxdx
23
Zxdx
3
Z 2 dx