Wallis Formula - Engineering Mathematics, Lecture notes of Engineering Mathematics

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WALLI’S FORMULA

TRIGONOMETRIC TRANSFORMATION

WALLI’S FORMULA

OBJECTIVES:

• recall and apply the different trigonometric

identities in transforming powers of sine and

cosine; and

• use Walli’s Formula to shorten the solution in

finding the antiderivative of powers of sine and

cosine

EXAMPLE Evaluate the following integrals.

¿

𝟐𝟏 𝝅

𝟓𝟏𝟐

¿

𝟏

𝟓

     

 10  8  6  4  2  2

1 7 5 3 1

1. 3 cos sin 3

2 2

0

8 



  

x x dx

    

     

1

10 8 6 4 2

2 6 4 2

2. 8 cos sin 8

2 3

0

7   

x x dx



𝟑.  𝟎

𝝅 𝟔

𝒄𝒐𝒔

𝟖 𝟑 𝒙 𝒅𝒙 ¿

𝟏

𝟑

 𝟎

𝝅 𝟐

𝒄𝒐𝒔

𝟖 𝒖 𝒅𝒖

¿

𝟏

𝟑

∙

( 𝟕 ) ( 𝟓 ) ( 𝟑 ) ( 𝟏 )

( 𝟖 ) ( 𝟔 ) ( 𝟒 ) ( 𝟐 )

∙

𝝅

𝟐

¿

𝟑𝟓 𝝅

𝟕𝟔𝟖

6 2

3 ; u 6

when x

when x 0 ; u 0

letu 3 x; du 3 dx

     

 

 