1000 Solved Problems in Modern Physics

(Grace) #1

1.3 Solutions 51


1.38 f(a+x)=sin(a+x)
Putx= 0
f(a)=sina
f′(a)=cosa
f′′(a)=−sina
f′′′(a)=−cosa
Substitute in


f(x)=f(a)+
(x−a)
1!

f′(a)+
(x−a)^2
2!

f′′(a)+
(x−a)^3
3!

f′′′(a)+···

sin(a+x)=sina+

x
1

cosa−

x^2
2!

sina−

x^3
3!

cosa+···

1.39 We know that


y= 1 +x+x^2 +x^3 +x^4 +··· = 1 /(1−x)
Differentiating with respect tox,
dy/dx= 1 + 2 x+ 3 x^2 + 4 x^3 +··· = 1 /(1−x)^2 =S

1.3.7 Integration........................................


1.40 (a)


sin^3 xcos^6 xdx=


sin^2 xcos^6 xsinxdx
=−


(1−cos^2 x) cos^6 xd(cosx)=


cos^8 xd(cosx)−


cos^6 xd(cosx)

=

cos^9 x
9


cos^7 x
7

+C

(b)


sin^4 xcos^2 xdx=


(sinxcosx)^2 sin^2 xdx

=


1

4

sin^22 x(

1

2


1

2

cos 2x)dx

=

1

8


sin^22 xdx−

1

8


sin^22 xcos 2xdx

=

1

8


(

1

2


1

2

cos 4x)dx−

1

8


sin^22 xcos 2xdx

=

x
16


sin 4x
64


sin^32 x
48

+C

1.41 Express the integrand as sum of functions.


Let
1
2 x^2 − 3 x− 2

=
1
(2x+1)(x−2)

=
A
2 x+ 1

+
B
x− 2

=
A(x−2)+B(2x+1)
(2x+1)(x−2)
B− 2 A= 1
A+ 2 B= 0
Solving,A=−^25 andB=^15

I=−

2

5


dx
2 x+ 1

+

1

5


dx
x− 2

+C
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