396 Higher Engineering Mathematics
- 3tan2t
[
3
2
ln(sec2t)+c
]
7.
2et
√
(et+ 4 )
[
4
√
(et+ 4 )+c
]
In Problems 8 to 10, evaluate the definite integrals
correct to 4 significant figures.
8.
∫ 1
0
3 xe(^2 x
(^2) − 1 )
dx [1.763]
9.
∫ π
2
0
3sin^4 θcosθdθ [0.6000]
10.
∫ 1
0
3 x
( 4 x^2 − 1 )^5
dx [0.09259]
- The electrostatic potential on all parts of a
conducting circular disc of radiusris given
by the equation:
V= 2 πσ
∫ 9
0
R
√
R^2 +r^2
dR
Solvetheequationbydeterminingtheintegral.
[
V= 2 πσ
{√
( 92 +r^2 )−r
}]
- In the study of a rigid rotor the following
integration occurs:
Zr=
∫∞
0
( 2 J+ 1 )e
−J(J+ 1 )h^2
8 π^2 Ik T dJ
Determine Zr for constant temperatureT
assumingh,Iandkare constants.
[
8 π^2 IkT
h^2
]
- In electrostatics,
E=
∫ π
0
⎧
⎨
⎩
a^2 σsinθ
2 ε
√(
a^2 −x^2 −2axcosθ
)dθ
⎫
⎬
⎭
where a,σandεare constants,xis greater
than a, andxis independent ofθ. Show that
E=
a^2 σ
εx