Find the derivative and simplify

Derivatives of the Arcsecant function and integrals that result in an

Arcsecant function.

Find the derivative and simplify.



1

(1) f ( x)  xarcsec 

x

(2) f ( x)  arcsec x 2  4

 ex  ex 

(3) f ( x)  arcsec

 



 2 

2 2x 3

(4) f ( x)  arcsec

3 3 3

 9x 2  1 

(5) f ( x)   x 2 arc sec 3 x   if x  1/ 3



 9 



(6) f ( x)  x arcsec(x)  ln x  x 2  1



Evaluate the Integrals







1

(1) dx

4x x2  9

2 3

1

(2)

2

 x x2  1

dx







tan x

(3) dx

4 cos 2 x  1

1

1

(4)

1

 x 4x2  1

dx

2







3

(5) dx

( x  2) x 2  4 x  3





10

(6) dt

(t  3) t 2  6t  16





6

(7) dt

(2t  1) 4t 2  4t  8





x

(8) dx

( x  5) x  10x  21

2 4 2

Derivatives of the Arcsecant function and integrals that result in an

Arcsecant function.







Solutions

Derivatives

1 x x

(1) f ' ( x)  arcsec   (2) f ' ( x) 

x 1 x2 ( x 2  4) ( x 2  3)

2 1

(3) f ' ( x)  x (4) f ' ( x) 

e  ex x 2x3  3

(5) 

f ' ( x)  2 x arc sec 3x 

2

(6) if x  1 , f ' ( x)  arcsec( ) if x  1 , f ' ( x)  arcsec( x) 

x

x2 1

Integrals

1 x 

(1) arcsec  C (2)

12 3 6



(3) arcsec2 cos x  C (4)

12

t 3

(5) 3arcsecx  2  C (6) 2arcsec C

5

2t  1 1 x2  5

(7) 3arcsec C (8) arcsec C

3 4 2