396 Higher Engineering Mathematics
6. 3 tan 2t
3
2
ln(sec 2t ) + c
7.
2e t
√
(e t + 4)
4
√
(e t + 4) + c
In Problems 8 to 10, evaluate the definite integrals
correct to 4 significant figures.
8.
1
0
3x e
(2x 2 −1) dx
[1.763]
9.
π
2
0
3 sin
4
θ cos θ dθ
[0.6000]
10.
1
0
3x
(4x 2 − 1) 5 dx
[0.09259]
11. The electrostatic potential on all parts of a
conducting circular disc of radius r is given
by the equation:
V = 2πσ
9
0
R
√
R 2 + r 2
d R
Solve the equation by determining the integral.
V = 2πσ
(9 2 + r 2 ) − r
12. In the study of a rigid rotor the following
integration occurs:
Z r =
∞
0
(2 J + 1)e
−J (J +1) h 2
8π 2 Ik T d J
Determine Z r for constant temperature T
assuming h, I and k are constants.
8π
2 I kT
h 2
13. In electrostatics,
E =
π
0
⎧
⎨
⎩
a 2 σsin θ
2ε
a 2 − x 2 − 2ax cos θ
dθ
⎫
⎬
⎭
where a, σ and ε are constants, x is greater
than a, and x is independent of θ. Show that
E =
a 2 σ
εx
6. 3 tan 2t
3
2
ln(sec 2t ) + c
7.
2e t
√
(e t + 4)
4
√
(e t + 4) + c
In Problems 8 to 10, evaluate the definite integrals
correct to 4 significant figures.
8.
1
0
3x e
(2x 2 −1) dx
[1.763]
9.
π
2
0
3 sin
4
θ cos θ dθ
[0.6000]
10.
1
0
3x
(4x 2 − 1) 5 dx
[0.09259]
11. The electrostatic potential on all parts of a
conducting circular disc of radius r is given
by the equation:
V = 2πσ
9
0
R
√
R 2 + r 2
d R
Solve the equation by determining the integral.
V = 2πσ
(9 2 + r 2 ) − r
12. In the study of a rigid rotor the following
integration occurs:
Z r =
∞
0
(2 J + 1)e
−J (J +1) h 2
8π 2 Ik T d J
Determine Z r for constant temperature T
assuming h, I and k are constants.
8π
2 I kT
h 2
13. In electrostatics,
E =
π
0
⎧
⎨
⎩
a 2 σsin θ
2ε
a 2 − x 2 − 2ax cos θ
dθ
⎫
⎬
⎭
where a, σ and ε are constants, x is greater
than a, and x is independent of θ. Show that
E =
a 2 σ
εx