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 Proof: Integral cosh(x) (Math | Calculus | Integrals | Table Of | cosh x)

 Discussion of cosh x dx = sinh x + C.

## 1. Proof

Strategy: Use definition of cosh.
 cosh x = ex + e-x  2
 cosh x dx = ex + e-x  2 dx
= (1/2)  ex dx + (1/2)  e-x dx
solve left equation
= (1/2) ex + (1/2)  e-x dx

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

substitute du= - dx, u= - x

= (1/2) ex - (1/2)  - e-x dx
= (1/2) ex - (1/2)  eu du

solve the right integral
= (1/2) ex - (1/2) eu + C

substitute back u= - x
= (1/2) ex - (1/2) e-x + C  = ex - e-x  2 + C
which by definition
= sinh x + C
Q.E.D.

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