Higher Engineering Mathematics, Sixth Edition

396 Higher Engineering Mathematics

6. 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]

11. 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

}]

12. 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

]

13. 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