Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(lu) #1

PRELIMINARY CALCULUS


Since

dt
dx

=

1
2

sec^2

x
2

=

1
2

(
1+tan^2

x
2

)
=

1+t^2
2

,

the required relationship is


dx=

2
1+t^2

dt. (2.34)

Evaluate the integral

I=


2


1+3cosx

dx.

Rewriting cosxin terms oftand using (2.34) yields


I=



2


1+3


[


(1−t^2 )(1 +t^2 )−^1

]


(


2


1+t^2

)


dt

=



2(1 +t^2 )
1+t^2 +3(1−t^2 )

(


2


1+t^2

)


dt

=



2


2 −t^2

dt=


2


(



2 −t)(


2+t)

dt

=



1



2


(


1



2 −t

+


1



2+t

)


dt

=−


1



2


ln(


2 −t)+

1



2


ln(


2+t)+c

=


1



2


ln

[√


2+tan(x/2)

2 −tan (x/2)

]


+c.

Integrals of a similar form to (2.33), but involving sin 2x,cos2x, tan 2x,sin^2 x,

cos^2 xor tan^2 xinstead of cosxand sinx, should be evaluated by using the


substitutiont=tanx.Inthiscase


sinx=

t

1+t^2

, cosx=

1

1+t^2

and dx=

dt
1+t^2

. (2.35)


A final example of the evaluation of integrals using substitution is the method

of completing the square (cf. subsection 1.7.3).

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