Dave's Math Tables: Integral sech(x)

(Math | Calculus | Integrals | Table Of | sech x)

Discussion of sech x = arctan (sinh x) + C.

1. Proof

Strategy: Use definition of csch; use algebra; use substitution.

sech x =

1  cosh x

=

2  ex + e-x

 

sech x dx =

2  ex + e-x

dx

multiply numerator and denominator by e^x

=

2  (ex + e-x)

(ex)  (ex)

dx

 

=

2 ex  (e2x + 1)

dx

set
  u = e^x
then we find
  du = e^x dx

substitute du = e^x dx, u = e^x
 

=

2 du  u2 + 1

= 2

du  u2 + 1


solve the integral

= 2 arctan(u) + C

substitute back u = e^x

= 2 arctan(e^x) + C
........
Q.E.D.