188 Chapter 6Methods of integration
- (u 1 = 1 sin 1 x) 28.
- (x 1 = 1 sin 1 θ)
31
- (i)Use the substitutionx 1 = 1 a 1 sinh 1 uto show that.
(ii)Use the substitution to show that
Evaluate the definite integrals:
38.Line shapes in magnetic resonance spectroscopy are often described by the Lorentz
function
.
Find
39.An approximate expression for the rotational partition function of a linear rotor is
whereθ
R
1 =1A
2
22 Ikis the rotational temperature,Iis the moment of inertia, and kis
Boltzmann’s constant. Evaluate the integral.
Section 6.4
Evaluate the integrals:
51.Z
0
2
3
π 2exdx
−xcos
Zebxdx
axZexdx cos
−xZ sin 2
0
1
2
xxdxln
Z
lnx
x
dx
2
Zxxdxln
Z0
22
∞
xe dx
−xZ
0
1
xe dx
xZxe dx
22 xZ()xxdx+ 1
2
Zxxdx cos 2
3
Zxxdxsin sin
qJe dJ
rJJ T
=+
−+
Z
0
1
21
∞
()
()θRZ
ωωω
0∞
gd().
g
T
T
()
()
ω
ωω
=
+−
1
1
2
0
2
π
Z
0
2∞
xe dx
−xZ
0
1
2
2
dx
−x
Z
0
π 2sin cosθθθd
Z
0
2πsin( x π)
x
dx
Z
1
2
2
32
xdx
x −
Z
dx
xa
xxa C
22
22
=++
ln +.
ux x a=+ +
22
Z
dx
xa
x
a
C
22
1
=
−
sinh
Z
x
x
dx u x
1 +
()=
Z
xdx
x
2
2
1 −
Z
dx
4 x
2
Zln cos sinxxdx
()Zsin
3
xxdxcos