Lecture 4: Techniques for Integration II. part II ...

Lecture 4: Techniques for Integration II. part II: Integral of the tan x, sec x

Recall

(tan x) = sec2 x

1 + tan2 x = sec2 x

(sec x) = sec x tan x

tan x dx = ln | sec x| + c = - ln | cos x| + c

ex. sec2 x dx =

, sec2 x tan3 x dx =

ex. sec x tan x dx =

, sec3 x tan x dx =

tan2 x dx, tan3 x sec x dx, sec x dx, sec3 x dx =?

1

tanm x secn xdx (1) n is even: Isolate a sec2 x ,rewrite the rest sec x in terms of tan x, if needed. Then use u-sub: let u = tan x. (2) m is odd, m > 1, n = 0: Isolate a tan x sec x , set the rest tan x in terms of sec x, if needed. Then use u-sub: let u = sec x. (3) m is even: use

tan2 x = sec2 x - 1, to convert the tanm x in terms of sec2 x.

NOTE: Similar rules apply to cotm x cscn xdx

2

Evaluate: ex. 1. tan6 x sec4 xdx (EVEN power of sec x )

(

tan7 7

x

+

tan9 x 9

+

c)

ex. 2. tan3 x sec xdx (ODD power of tan x )

(

sec3 3

x

-

sec x

+

c)

3

ex. 3. tan2 xdx (EVEN power of tan x )

(tan x - x + c)

ex. 4. sec xdx (ln | sec x + tan x| + c)

4

ex. 5.

sec3 xdx

(

1 2

(sec

x

tan

x

+

ln

|

sec

x

+

tan

x|)

+

c)

NOTE: In this class, we will learn how to evaluate the integrals of the sec x, tan x and csc x, cot x without using a table of integrals.

5

NYTI Integration gymnastics:

1. Similarly, you can apply the same technique here to evaluate the integrals of the cosecant and cotangent functions.

(cot x) = - csc2 x, 1+cot2 x = csc2 x (csc x) = - csc x cot x

cot x dx = ln | sin x| + c

ex. Evaluate csc2 x dx =

, csc2 x cot3 x dx =

ex. Evaluate csc x cot x dx =

, csc3 x cot x dx =

ex. Evaluate csc xdx (ln | csc x - cot x| + c)

ex. Evaluate csc3 xdx

6

2.

tan4 xdx

(

tan3 3

x

-

tan

x

+

x

+

c)

3. tan3 xdx Idea:

(

1 2

tan2

x

+

ln

|

cos

x|

+

c)

4.

tan xdx 5

tan4 x 4

-

tan2 2

x

-

ln| cos x|

+

c

cos5 x

5.

sin xdx 2 sin x

-

4 sin5/2 5

x

+

2 sin9/2 x 9

+

c

6. x sec x tan x dx x sec x - ln| sec x + tan x| + c

cos x + sin x

7.

dx

sin 2x

7

1 - tan2 x

8.

sec2 x dx

9. x cos2 x dx

10. cos x cos5(sin x) dx

tan3 x

tan4 x tan6 x

11. Show cos4 x dx = 4 + 6 + c

dx 12.

cos x - 1

13. x sec2(x2) tan4(x2) dx

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