Math2.org Math Tables: Integral sinh(x)

Discussion of sinh x dx = cosh x + C.

1. Proof

Strategy: Use definition of sinh.
 

sinh x =

ex - e-x 2

sinh x dx =

ex - e-x 2

dx

= (1/2)  e^x dx - (1/2)  e^-x dx
solve left equation
= (1/2) e^x - (1/2)  e^-x dx

set
  u = - x
then we find
  du = - dx

substitute du= - dx, u= - x

= (1/2) e^x + (1/2)  - e^-x dx
= (1/2) e^x + (1/2)  e^u du

solve the right integral
= (1/2) e^x + (1/2) e^u + C

substitute back u= - x
= (1/2) e^x + (1/2) e^-x + C

=

ex + e-x 2

+ C

which by definition
= cosh x + C
Q.E.D.