Integral cosh(x)

Themes > Science > Mathematics > Calculus > Integrals > Integrals Table > Integral cosh(x)

Discussion of (integral)

cosh x dx = sinh x + C.

1. Proof

Strategy: Use definition of cosh.

cosh x =

e^x + e^-x2

cosh x dx = (integral)

e^x + e^-x2

= (1/2)

e^x dx + (1/2) (integral)

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

e^-x dx

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

substitute du= - dx, u= - x

= (1/2) e^x - (1/2) (integral) - 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

= e^x - e^-x2 + C
which by definition
= sinh x + C
Q.E.D.

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