Integration of Trigonometric Functions: A Comprehensive Guide with Examples and Exercises, Slides of Differential and Integral Calculus

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ANTIDERIVATIVES

(INTEGRAL)

TRANSFORMATION OF

TRIGONOMETRIC FUNCTIONS

OBJECTIVES:

• integrate functions of the nth power of

the different trigonometric functions;

and

• correlate the basic trigonometric

integration formulas in evaluating the

integral of the nth power of

trigonometric functions.

Part 1: Powers of Sine and Cosine

usetheidentity:sin cos 1 sin or cos wheren isanypositiveodd integer 1 :Consider the integrand n n n     u u udu u du Case n   u  u  u u u du u du Case 1 cos 2 2

cos 1 cos 2 2

usetheidentity: sin sin or cos wheren isanypositiveeven integer 2 :Consider the integrand n n n n  

 

Part 1: Powers of Sine and Cosine

sametechniqueasthatof Case 1. oneof themor nisapositiveoddinteger, employ the functionsof theform sin cos du whereat least 3 :If theintergrandcontainstheproductof sineand cosine n  u  u Case m positiveeveninteger,employ thesametechniqueasthatof Case 2. functionsof theform sin cos du whereboth mor nis a 4 :If theintergrandcontainstheproductof sineand cosine n  u  u Case m

EXAMPLE: Evaluate the following.    x dx x 5 2 sin ln 1

 x xdx 4 4

2. sin cos   dx x x x cos 3 sin 3 cos 3

8 4 4    x  x dx 2

4. sin 3 cos 5   dx x x x 2 sin 2 cos 2 cos

6 3 2

CLASSWORK: Evaluate the following.

 dx x x sin 2 1 sin 2

8 4 2

dx

x x

x

sin 2 cos 2

cos 4

dx

x

x

csc 4 1

log sec 4

2 4

x  x dx sec 3 1 cos 3

2

EXAMPLE: Evaluate the following.

x d x



1. tan

4 

2. cot 2 xdx 3   
3. sec x  1 dx 4 i.   
4. sec 2 x dx 3   
5. csc 2 x dx 2   
6. csc 2 x  1 dx

x x d x



7. tan sec

2 2

x x d x



8. tan sec

3 3

x x d x



9. tan sec

3 2

x x d x



10. tan sec

2 3

HOMEWORK: Evaluate the following. 

dx

x

x x

cos 2

cos 2 sin 2

6 2 3    x  x dx 2 2

2. tan 2 cot 2      
3. cos 4 x sin sin 4 x dx 3 i.  dx x x tan 3 4

lnsin 3     8 2 2 sec 1

x xdx  dx x x   sec tan

3