Mathematical Methods for Physics and Engineering : A Comprehensive Guide

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).